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10:03:08",{"id":1084,"title":1085,"abbrlink":1086,"body":1087,"category":40,"cover":156,"date":156,"description":1093,"extension":8953,"mathjax":8954,"meta":8955,"navigation":8954,"path":862,"published":864,"readingMinutes":8958,"seo":8959,"stem":8960,"sticky":156,"swiper_index":156,"tags":8961,"updated":156,"words":863,"__hash__":8962},"posts\u002Fposts\u002F数据结构\u002F2025-09-29-数据结构-图.md","数据结构-图","18076",{"type":1088,"value":1089,"toc":8934},"minimark",[1090,1094,1097,1100,1108,1631,1638,1641,1761,2029,2040,2059,2219,2225,2242,2248,2251,2730,2735,4578,4582,5866,5871,6533,6538,6543,6548,6553,6558,6564,6575,6595,6602,6608,6698,7204,7207,7212,7215,7691,7776,7781,7784,7792,7798,7801,7815,7818,7824,7933,7936,7942,7945,8035,8038,8173,8179,8185,8188,8191,8194,8197,8202,8205,8210,8493,8499,8502,8508,8514,8517,8667,8669,8675,8744,8817,8820,8826,8829,8840,8843,8851,8857,8860,8865,8868,8871,8874,8879,8882,8885,8891,8894,8900,8903,8908,8911,8914,8917,8922,8925,8928],[1091,1092,1093],"p",{},"图是一种比线性表和树更为复杂的数据结构。在图结构中，结点之间的关系可以是任意的，图中任意两个数据元素之间都可能相关。",[1091,1095,1096],{},"图结构在计算机科学和算法设计中有广泛的应用。例如，在社交网络分析中，可以使用图结构来表示用户之间的关系；在路线规划中，可以使用图结构来表示道路网络和城市之间的连接关系；在人工智能领域中，图结构可以用于表示知识图谱和推荐系统等。",[1091,1098,1099],{},"在离散数学中，图论是专门研究图的性质的数学分支，而在数据结构中，则应用图论的知识讨论如何在计算机上实现图的操作，因此主要学习图的存储结构，以及若干图的操作的实现。",[1101,1102,1104],"h2",{"id":1103},"一定义",[1105,1106,1107],"strong",{},"一、定义",[1091,1109,1110,1111,1157,1158,1188,1189,1219,1220,1308,1309,1337,1338,1366,1367,1395,1396,1188,1440,1484,1485,1513,1514,1557,1558,1601,1602,1630],{},"图（Graph）",[1112,1113,1116,1138],"span",{"className":1114},[1115],"katex",[1112,1117,1120],{"className":1118},[1119],"katex-mathml",[1121,1122,1124],"math",{"xmlns":1123},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[1125,1126,1127,1134],"semantics",{},[1128,1129,1130],"mrow",{},[1131,1132,1133],"mi",{},"G",[1135,1136,1133],"annotation",{"encoding":1137},"application\u002Fx-tex",[1112,1139,1143],{"className":1140,"ariaHidden":1142},[1141],"katex-html","true",[1112,1144,1147,1152],{"className":1145},[1146],"base",[1112,1148],{"className":1149,"style":1151},[1150],"strut","height:0.6833em;",[1112,1153,1133],{"className":1154},[1155,1156],"mord","mathnormal","由两个集合",[1112,1159,1161,1175],{"className":1160},[1115],[1112,1162,1164],{"className":1163},[1119],[1121,1165,1166],{"xmlns":1123},[1125,1167,1168,1173],{},[1128,1169,1170],{},[1131,1171,1172],{},"V",[1135,1174,1172],{"encoding":1137},[1112,1176,1178],{"className":1177,"ariaHidden":1142},[1141],[1112,1179,1181,1184],{"className":1180},[1146],[1112,1182],{"className":1183,"style":1151},[1150],[1112,1185,1172],{"className":1186,"style":1187},[1155,1156],"margin-right:0.2222em;","和",[1112,1190,1192,1206],{"className":1191},[1115],[1112,1193,1195],{"className":1194},[1119],[1121,1196,1197],{"xmlns":1123},[1125,1198,1199,1204],{},[1128,1200,1201],{},[1131,1202,1203],{},"E",[1135,1205,1203],{"encoding":1137},[1112,1207,1209],{"className":1208,"ariaHidden":1142},[1141],[1112,1210,1212,1215],{"className":1211},[1146],[1112,1213],{"className":1214,"style":1151},[1150],[1112,1216,1203],{"className":1217,"style":1218},[1155,1156],"margin-right:0.0576em;","组成，记为",[1112,1221,1223,1255],{"className":1222},[1115],[1112,1224,1226],{"className":1225},[1119],[1121,1227,1228],{"xmlns":1123},[1125,1229,1230,1252],{},[1128,1231,1232,1234,1238,1242,1244,1247,1249],{},[1131,1233,1133],{},[1235,1236,1237],"mo",{},"=",[1235,1239,1241],{"stretchy":1240},"false","(",[1131,1243,1172],{},[1235,1245,1246],{"separator":1142},",",[1131,1248,1203],{},[1235,1250,1251],{"stretchy":1240},")",[1135,1253,1254],{"encoding":1137},"G=(V, E)",[1112,1256,1258,1279],{"className":1257,"ariaHidden":1142},[1141],[1112,1259,1261,1264,1267,1272,1276],{"className":1260},[1146],[1112,1262],{"className":1263,"style":1151},[1150],[1112,1265,1133],{"className":1266},[1155,1156],[1112,1268],{"className":1269,"style":1271},[1270],"mspace","margin-right:0.2778em;",[1112,1273,1237],{"className":1274},[1275],"mrel",[1112,1277],{"className":1278,"style":1271},[1270],[1112,1280,1282,1286,1290,1293,1297,1301,1304],{"className":1281},[1146],[1112,1283],{"className":1284,"style":1285},[1150],"height:1em;vertical-align:-0.25em;",[1112,1287,1241],{"className":1288},[1289],"mopen",[1112,1291,1172],{"className":1292,"style":1187},[1155,1156],[1112,1294,1246],{"className":1295},[1296],"mpunct",[1112,1298],{"className":1299,"style":1300},[1270],"margin-right:0.1667em;",[1112,1302,1203],{"className":1303,"style":1218},[1155,1156],[1112,1305,1251],{"className":1306},[1307],"mclose","，其中",[1112,1310,1312,1325],{"className":1311},[1115],[1112,1313,1315],{"className":1314},[1119],[1121,1316,1317],{"xmlns":1123},[1125,1318,1319,1323],{},[1128,1320,1321],{},[1131,1322,1172],{},[1135,1324,1172],{"encoding":1137},[1112,1326,1328],{"className":1327,"ariaHidden":1142},[1141],[1112,1329,1331,1334],{"className":1330},[1146],[1112,1332],{"className":1333,"style":1151},[1150],[1112,1335,1172],{"className":1336,"style":1187},[1155,1156],"是顶点的有穷非空集合，",[1112,1339,1341,1354],{"className":1340},[1115],[1112,1342,1344],{"className":1343},[1119],[1121,1345,1346],{"xmlns":1123},[1125,1347,1348,1352],{},[1128,1349,1350],{},[1131,1351,1203],{},[1135,1353,1203],{"encoding":1137},[1112,1355,1357],{"className":1356,"ariaHidden":1142},[1141],[1112,1358,1360,1363],{"className":1359},[1146],[1112,1361],{"className":1362,"style":1151},[1150],[1112,1364,1203],{"className":1365,"style":1218},[1155,1156],"是",[1112,1368,1370,1383],{"className":1369},[1115],[1112,1371,1373],{"className":1372},[1119],[1121,1374,1375],{"xmlns":1123},[1125,1376,1377,1381],{},[1128,1378,1379],{},[1131,1380,1172],{},[1135,1382,1172],{"encoding":1137},[1112,1384,1386],{"className":1385,"ariaHidden":1142},[1141],[1112,1387,1389,1392],{"className":1388},[1146],[1112,1390],{"className":1391,"style":1151},[1150],[1112,1393,1172],{"className":1394,"style":1187},[1155,1156],"中顶点偶对的有穷集合，这些顶点偶对称为边。\n",[1112,1397,1399,1419],{"className":1398},[1115],[1112,1400,1402],{"className":1401},[1119],[1121,1403,1404],{"xmlns":1123},[1125,1405,1406,1416],{},[1128,1407,1408,1410,1412,1414],{},[1131,1409,1172],{},[1235,1411,1241],{"stretchy":1240},[1131,1413,1133],{},[1235,1415,1251],{"stretchy":1240},[1135,1417,1418],{"encoding":1137},"V(G)",[1112,1420,1422],{"className":1421,"ariaHidden":1142},[1141],[1112,1423,1425,1428,1431,1434,1437],{"className":1424},[1146],[1112,1426],{"className":1427,"style":1285},[1150],[1112,1429,1172],{"className":1430,"style":1187},[1155,1156],[1112,1432,1241],{"className":1433},[1289],[1112,1435,1133],{"className":1436},[1155,1156],[1112,1438,1251],{"className":1439},[1307],[1112,1441,1443,1463],{"className":1442},[1115],[1112,1444,1446],{"className":1445},[1119],[1121,1447,1448],{"xmlns":1123},[1125,1449,1450,1460],{},[1128,1451,1452,1454,1456,1458],{},[1131,1453,1203],{},[1235,1455,1241],{"stretchy":1240},[1131,1457,1133],{},[1235,1459,1251],{"stretchy":1240},[1135,1461,1462],{"encoding":1137},"E(G)",[1112,1464,1466],{"className":1465,"ariaHidden":1142},[1141],[1112,1467,1469,1472,1475,1478,1481],{"className":1468},[1146],[1112,1470],{"className":1471,"style":1285},[1150],[1112,1473,1203],{"className":1474,"style":1218},[1155,1156],[1112,1476,1241],{"className":1477},[1289],[1112,1479,1133],{"className":1480},[1155,1156],[1112,1482,1251],{"className":1483},[1307],"通常分别表示图",[1112,1486,1488,1501],{"className":1487},[1115],[1112,1489,1491],{"className":1490},[1119],[1121,1492,1493],{"xmlns":1123},[1125,1494,1495,1499],{},[1128,1496,1497],{},[1131,1498,1133],{},[1135,1500,1133],{"encoding":1137},[1112,1502,1504],{"className":1503,"ariaHidden":1142},[1141],[1112,1505,1507,1510],{"className":1506},[1146],[1112,1508],{"className":1509,"style":1151},[1150],[1112,1511,1133],{"className":1512},[1155,1156],"的顶点集合和边集合，",[1112,1515,1517,1536],{"className":1516},[1115],[1112,1518,1520],{"className":1519},[1119],[1121,1521,1522],{"xmlns":1123},[1125,1523,1524,1534],{},[1128,1525,1526,1528,1530,1532],{},[1131,1527,1203],{},[1235,1529,1241],{"stretchy":1240},[1131,1531,1133],{},[1235,1533,1251],{"stretchy":1240},[1135,1535,1462],{"encoding":1137},[1112,1537,1539],{"className":1538,"ariaHidden":1142},[1141],[1112,1540,1542,1545,1548,1551,1554],{"className":1541},[1146],[1112,1543],{"className":1544,"style":1285},[1150],[1112,1546,1203],{"className":1547,"style":1218},[1155,1156],[1112,1549,1241],{"className":1550},[1289],[1112,1552,1133],{"className":1553},[1155,1156],[1112,1555,1251],{"className":1556},[1307],"可以为空集。\n若",[1112,1559,1561,1580],{"className":1560},[1115],[1112,1562,1564],{"className":1563},[1119],[1121,1565,1566],{"xmlns":1123},[1125,1567,1568,1578],{},[1128,1569,1570,1572,1574,1576],{},[1131,1571,1203],{},[1235,1573,1241],{"stretchy":1240},[1131,1575,1133],{},[1235,1577,1251],{"stretchy":1240},[1135,1579,1462],{"encoding":1137},[1112,1581,1583],{"className":1582,"ariaHidden":1142},[1141],[1112,1584,1586,1589,1592,1595,1598],{"className":1585},[1146],[1112,1587],{"className":1588,"style":1285},[1150],[1112,1590,1203],{"className":1591,"style":1218},[1155,1156],[1112,1593,1241],{"className":1594},[1289],[1112,1596,1133],{"className":1597},[1155,1156],[1112,1599,1251],{"className":1600},[1307],"为空，则图",[1112,1603,1605,1618],{"className":1604},[1115],[1112,1606,1608],{"className":1607},[1119],[1121,1609,1610],{"xmlns":1123},[1125,1611,1612,1616],{},[1128,1613,1614],{},[1131,1615,1133],{},[1135,1617,1133],{"encoding":1137},[1112,1619,1621],{"className":1620,"ariaHidden":1142},[1141],[1112,1622,1624,1627],{"className":1623},[1146],[1112,1625],{"className":1626,"style":1151},[1150],[1112,1628,1133],{"className":1629},[1155,1156],"只有顶点而没有边。",[1091,1632,1633],{},[1634,1635],"img",{"alt":1636,"src":1637},"alt text","https:\u002F\u002Fpica.zhimg.com\u002F80\u002Fv2-375dc00a65c91200298544017c365a46_720w.webp",[1091,1639,1640],{},"如上图所示，左图为有向图，右图为无向图。",[1091,1642,1643,1644,1672,1673,1716,1717,1760],{},"对于图",[1112,1645,1647,1660],{"className":1646},[1115],[1112,1648,1650],{"className":1649},[1119],[1121,1651,1652],{"xmlns":1123},[1125,1653,1654,1658],{},[1128,1655,1656],{},[1131,1657,1133],{},[1135,1659,1133],{"encoding":1137},[1112,1661,1663],{"className":1662,"ariaHidden":1142},[1141],[1112,1664,1666,1669],{"className":1665},[1146],[1112,1667],{"className":1668,"style":1151},[1150],[1112,1670,1133],{"className":1671},[1155,1156],"，若边集",[1112,1674,1676,1695],{"className":1675},[1115],[1112,1677,1679],{"className":1678},[1119],[1121,1680,1681],{"xmlns":1123},[1125,1682,1683,1693],{},[1128,1684,1685,1687,1689,1691],{},[1131,1686,1203],{},[1235,1688,1241],{"stretchy":1240},[1131,1690,1133],{},[1235,1692,1251],{"stretchy":1240},[1135,1694,1462],{"encoding":1137},[1112,1696,1698],{"className":1697,"ariaHidden":1142},[1141],[1112,1699,1701,1704,1707,1710,1713],{"className":1700},[1146],[1112,1702],{"className":1703,"style":1285},[1150],[1112,1705,1203],{"className":1706,"style":1218},[1155,1156],[1112,1708,1241],{"className":1709},[1289],[1112,1711,1133],{"className":1712},[1155,1156],[1112,1714,1251],{"className":1715},[1307],"为有向边的集合，则称该图为有向图；\n若边集",[1112,1718,1720,1739],{"className":1719},[1115],[1112,1721,1723],{"className":1722},[1119],[1121,1724,1725],{"xmlns":1123},[1125,1726,1727,1737],{},[1128,1728,1729,1731,1733,1735],{},[1131,1730,1203],{},[1235,1732,1241],{"stretchy":1240},[1131,1734,1133],{},[1235,1736,1251],{"stretchy":1240},[1135,1738,1462],{"encoding":1137},[1112,1740,1742],{"className":1741,"ariaHidden":1142},[1141],[1112,1743,1745,1748,1751,1754,1757],{"className":1744},[1146],[1112,1746],{"className":1747,"style":1285},[1150],[1112,1749,1203],{"className":1750,"style":1218},[1155,1156],[1112,1752,1241],{"className":1753},[1289],[1112,1755,1133],{"className":1756},[1155,1156],[1112,1758,1251],{"className":1759},[1307],"为无向边的集合，则称该图为无向图。",[1091,1762,1763,1764,1835,1836,1899,1900,1964,1965,2028],{},"在有向图中，顶点对",[1112,1765,1767,1793],{"className":1766},[1115],[1112,1768,1770],{"className":1769},[1119],[1121,1771,1772],{"xmlns":1123},[1125,1773,1774,1790],{},[1128,1775,1776,1779,1782,1784,1787],{},[1235,1777,1778],{},"\u003C",[1131,1780,1781],{},"x",[1235,1783,1246],{"separator":1142},[1131,1785,1786],{},"y",[1235,1788,1789],{},">",[1135,1791,1792],{"encoding":1137},"\u003Cx, y>",[1112,1794,1796,1809],{"className":1795,"ariaHidden":1142},[1141],[1112,1797,1799,1803,1806],{"className":1798},[1146],[1112,1800],{"className":1801,"style":1802},[1150],"height:0.5782em;vertical-align:-0.0391em;",[1112,1804,1778],{"className":1805},[1275],[1112,1807],{"className":1808,"style":1271},[1270],[1112,1810,1812,1816,1819,1822,1825,1829,1832],{"className":1811},[1146],[1112,1813],{"className":1814,"style":1815},[1150],"height:0.7335em;vertical-align:-0.1944em;",[1112,1817,1781],{"className":1818},[1155,1156],[1112,1820,1246],{"className":1821},[1296],[1112,1823],{"className":1824,"style":1300},[1270],[1112,1826,1786],{"className":1827,"style":1828},[1155,1156],"margin-right:0.0359em;",[1112,1830],{"className":1831,"style":1271},[1270],[1112,1833,1789],{"className":1834},[1275],"是有序的，它称为从顶点 x 到顶点 y 的一条有向边。因此，",[1112,1837,1839,1860],{"className":1838},[1115],[1112,1840,1842],{"className":1841},[1119],[1121,1843,1844],{"xmlns":1123},[1125,1845,1846,1858],{},[1128,1847,1848,1850,1852,1854,1856],{},[1235,1849,1778],{},[1131,1851,1781],{},[1235,1853,1246],{"separator":1142},[1131,1855,1786],{},[1235,1857,1789],{},[1135,1859,1792],{"encoding":1137},[1112,1861,1863,1875],{"className":1862,"ariaHidden":1142},[1141],[1112,1864,1866,1869,1872],{"className":1865},[1146],[1112,1867],{"className":1868,"style":1802},[1150],[1112,1870,1778],{"className":1871},[1275],[1112,1873],{"className":1874,"style":1271},[1270],[1112,1876,1878,1881,1884,1887,1890,1893,1896],{"className":1877},[1146],[1112,1879],{"className":1880,"style":1815},[1150],[1112,1882,1781],{"className":1883},[1155,1156],[1112,1885,1246],{"className":1886},[1296],[1112,1888],{"className":1889,"style":1300},[1270],[1112,1891,1786],{"className":1892,"style":1828},[1155,1156],[1112,1894],{"className":1895,"style":1271},[1270],[1112,1897,1789],{"className":1898},[1275],"与",[1112,1901,1903,1925],{"className":1902},[1115],[1112,1904,1906],{"className":1905},[1119],[1121,1907,1908],{"xmlns":1123},[1125,1909,1910,1922],{},[1128,1911,1912,1914,1916,1918,1920],{},[1235,1913,1778],{},[1131,1915,1786],{},[1235,1917,1246],{"separator":1142},[1131,1919,1781],{},[1235,1921,1789],{},[1135,1923,1924],{"encoding":1137},"\u003Cy, x>",[1112,1926,1928,1940],{"className":1927,"ariaHidden":1142},[1141],[1112,1929,1931,1934,1937],{"className":1930},[1146],[1112,1932],{"className":1933,"style":1802},[1150],[1112,1935,1778],{"className":1936},[1275],[1112,1938],{"className":1939,"style":1271},[1270],[1112,1941,1943,1946,1949,1952,1955,1958,1961],{"className":1942},[1146],[1112,1944],{"className":1945,"style":1815},[1150],[1112,1947,1786],{"className":1948,"style":1828},[1155,1156],[1112,1950,1246],{"className":1951},[1296],[1112,1953],{"className":1954,"style":1300},[1270],[1112,1956,1781],{"className":1957},[1155,1156],[1112,1959],{"className":1960,"style":1271},[1270],[1112,1962,1789],{"className":1963},[1275],"是不同的两条边。顶点对用一对尖括号括起来，x 是有向边的始点，y 是有向边的终点。",[1112,1966,1968,1989],{"className":1967},[1115],[1112,1969,1971],{"className":1970},[1119],[1121,1972,1973],{"xmlns":1123},[1125,1974,1975,1987],{},[1128,1976,1977,1979,1981,1983,1985],{},[1235,1978,1778],{},[1131,1980,1781],{},[1235,1982,1246],{"separator":1142},[1131,1984,1786],{},[1235,1986,1789],{},[1135,1988,1792],{"encoding":1137},[1112,1990,1992,2004],{"className":1991,"ariaHidden":1142},[1141],[1112,1993,1995,1998,2001],{"className":1994},[1146],[1112,1996],{"className":1997,"style":1802},[1150],[1112,1999,1778],{"className":2000},[1275],[1112,2002],{"className":2003,"style":1271},[1270],[1112,2005,2007,2010,2013,2016,2019,2022,2025],{"className":2006},[1146],[1112,2008],{"className":2009,"style":1815},[1150],[1112,2011,1781],{"className":2012},[1155,1156],[1112,2014,1246],{"className":2015},[1296],[1112,2017],{"className":2018,"style":1300},[1270],[1112,2020,1786],{"className":2021,"style":1828},[1155,1156],[1112,2023],{"className":2024,"style":1271},[1270],[1112,2026,1789],{"className":2027},[1275],"也称作一条弧，其中 x 为弧尾，y 为弧头。",[2030,2031,2036],"pre",{"className":2032,"code":2034,"language":2035},[2033],"language-text","A ──→ B\n│     │\n↓     ↓\nC ──→ D\n","text",[2037,2038,2034],"code",{"__ignoreMap":2039},"",[2041,2042,2043,2047,2050,2053,2056],"ul",{},[2044,2045,2046],"li",{},"\u003CA,B> 表示从顶点 A 到顶点 B 的有向边",[2044,2048,2049],{},"\u003CA,C> 表示从顶点 A 到顶点 C 的有向边",[2044,2051,2052],{},"\u003CB,D> 表示从顶点 B 到顶点 D 的有向边",[2044,2054,2055],{},"\u003CC,D> 表示从顶点 C 到顶点 D 的有向边",[2044,2057,2058],{},"箭头表示方向，\u003CA,B> ≠ \u003CB,A>",[1091,2060,2061,2062,2114,2115,1899,2166,2218],{},"在无向图中，顶点对",[1112,2063,2065,2087],{"className":2064},[1115],[1112,2066,2068],{"className":2067},[1119],[1121,2069,2070],{"xmlns":1123},[1125,2071,2072,2084],{},[1128,2073,2074,2076,2078,2080,2082],{},[1235,2075,1241],{"stretchy":1240},[1131,2077,1781],{},[1235,2079,1246],{"separator":1142},[1131,2081,1786],{},[1235,2083,1251],{"stretchy":1240},[1135,2085,2086],{"encoding":1137},"(x, y)",[1112,2088,2090],{"className":2089,"ariaHidden":1142},[1141],[1112,2091,2093,2096,2099,2102,2105,2108,2111],{"className":2092},[1146],[1112,2094],{"className":2095,"style":1285},[1150],[1112,2097,1241],{"className":2098},[1289],[1112,2100,1781],{"className":2101},[1155,1156],[1112,2103,1246],{"className":2104},[1296],[1112,2106],{"className":2107,"style":1300},[1270],[1112,2109,1786],{"className":2110,"style":1828},[1155,1156],[1112,2112,1251],{"className":2113},[1307],"是无序的，它称为与顶点 x 和顶点 y 相关联的一条边。这条边没有特定的方向，",[1112,2116,2118,2139],{"className":2117},[1115],[1112,2119,2121],{"className":2120},[1119],[1121,2122,2123],{"xmlns":1123},[1125,2124,2125,2137],{},[1128,2126,2127,2129,2131,2133,2135],{},[1235,2128,1241],{"stretchy":1240},[1131,2130,1781],{},[1235,2132,1246],{"separator":1142},[1131,2134,1786],{},[1235,2136,1251],{"stretchy":1240},[1135,2138,2086],{"encoding":1137},[1112,2140,2142],{"className":2141,"ariaHidden":1142},[1141],[1112,2143,2145,2148,2151,2154,2157,2160,2163],{"className":2144},[1146],[1112,2146],{"className":2147,"style":1285},[1150],[1112,2149,1241],{"className":2150},[1289],[1112,2152,1781],{"className":2153},[1155,1156],[1112,2155,1246],{"className":2156},[1296],[1112,2158],{"className":2159,"style":1300},[1270],[1112,2161,1786],{"className":2162,"style":1828},[1155,1156],[1112,2164,1251],{"className":2165},[1307],[1112,2167,2169,2191],{"className":2168},[1115],[1112,2170,2172],{"className":2171},[1119],[1121,2173,2174],{"xmlns":1123},[1125,2175,2176,2188],{},[1128,2177,2178,2180,2182,2184,2186],{},[1235,2179,1241],{"stretchy":1240},[1131,2181,1786],{},[1235,2183,1246],{"separator":1142},[1131,2185,1781],{},[1235,2187,1251],{"stretchy":1240},[1135,2189,2190],{"encoding":1137},"(y, x)",[1112,2192,2194],{"className":2193,"ariaHidden":1142},[1141],[1112,2195,2197,2200,2203,2206,2209,2212,2215],{"className":2196},[1146],[1112,2198],{"className":2199,"style":1285},[1150],[1112,2201,1241],{"className":2202},[1289],[1112,2204,1786],{"className":2205,"style":1828},[1155,1156],[1112,2207,1246],{"className":2208},[1296],[1112,2210],{"className":2211,"style":1300},[1270],[1112,2213,1781],{"className":2214},[1155,1156],[1112,2216,1251],{"className":2217},[1307],"是同一条边。为了有别于有向图，无向图的顶点对用一对圆括号括起来。",[2030,2220,2223],{"className":2221,"code":2222,"language":2035},[2033],"A ---- B\n|      |\n|      |\nC ---- D\n",[2037,2224,2222],{"__ignoreMap":2039},[2041,2226,2227,2230,2233,2236,2239],{},[2044,2228,2229],{},"(A,B) 表示顶点 A 和顶点 B 之间的无向边",[2044,2231,2232],{},"(A,C) 表示顶点 A 和顶点 C 之间的无向边",[2044,2234,2235],{},"(B,D) 表示顶点 B 和顶点 D 之间的无向边",[2044,2237,2238],{},"(C,D) 表示顶点 C 和顶点 D 之间的无向边",[2044,2240,2241],{},"没有方向性，(A,B) = (B,A)",[1101,2243,2245],{"id":2244},"二基本术语",[1105,2246,2247],{},"二、基本术语",[1091,2249,2250],{},"下面介绍图结构中的一些基本术语（注：n 表示图中顶点数目，e 表示边的数目）。",[2041,2252,2253],{},[2044,2254,2255,2256,2329,2330,2520,2521,2580,2581,2637,2638,2700,2701,2729],{},"子图：假设右两个图 ",[1112,2257,2259,2284],{"className":2258},[1115],[1112,2260,2262],{"className":2261},[1119],[1121,2263,2264],{"xmlns":1123},[1125,2265,2266,2282],{},[1128,2267,2268,2270,2272,2274,2276,2278,2280],{},[1131,2269,1133],{},[1235,2271,1237],{},[1235,2273,1241],{"stretchy":1240},[1131,2275,1172],{},[1235,2277,1246],{"separator":1142},[1131,2279,1203],{},[1235,2281,1251],{"stretchy":1240},[1135,2283,1254],{"encoding":1137},[1112,2285,2287,2305],{"className":2286,"ariaHidden":1142},[1141],[1112,2288,2290,2293,2296,2299,2302],{"className":2289},[1146],[1112,2291],{"className":2292,"style":1151},[1150],[1112,2294,1133],{"className":2295},[1155,1156],[1112,2297],{"className":2298,"style":1271},[1270],[1112,2300,1237],{"className":2301},[1275],[1112,2303],{"className":2304,"style":1271},[1270],[1112,2306,2308,2311,2314,2317,2320,2323,2326],{"className":2307},[1146],[1112,2309],{"className":2310,"style":1285},[1150],[1112,2312,1241],{"className":2313},[1289],[1112,2315,1172],{"className":2316,"style":1187},[1155,1156],[1112,2318,1246],{"className":2319},[1296],[1112,2321],{"className":2322,"style":1300},[1270],[1112,2324,1203],{"className":2325,"style":1218},[1155,1156],[1112,2327,1251],{"className":2328},[1307]," 和 ",[1112,2331,2333,2375],{"className":2332},[1115],[1112,2334,2336],{"className":2335},[1119],[1121,2337,2338],{"xmlns":1123},[1125,2339,2340,2372],{},[1128,2341,2342,2352,2354,2356,2362,2364,2370],{},[2343,2344,2345,2347],"msup",{},[1131,2346,1133],{},[1235,2348,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},"normal","0em","′",[1235,2353,1237],{},[1235,2355,1241],{"stretchy":1240},[2343,2357,2358,2360],{},[1131,2359,1172],{},[1235,2361,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1235,2363,1246],{"separator":1142},[2343,2365,2366,2368],{},[1131,2367,1203],{},[1235,2369,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1235,2371,1251],{"stretchy":1240},[1135,2373,2374],{"encoding":1137},"G'=(V', E')",[1112,2376,2378,2437],{"className":2377,"ariaHidden":1142},[1141],[1112,2379,2381,2385,2428,2431,2434],{"className":2380},[1146],[1112,2382],{"className":2383,"style":2384},[1150],"height:0.7519em;",[1112,2386,2388,2391],{"className":2387},[1155],[1112,2389,1133],{"className":2390},[1155,1156],[1112,2392,2395],{"className":2393},[2394],"msupsub",[1112,2396,2399],{"className":2397},[2398],"vlist-t",[1112,2400,2403],{"className":2401},[2402],"vlist-r",[1112,2404,2407],{"className":2405,"style":2384},[2406],"vlist",[1112,2408,2410,2415],{"style":2409},"top:-3.063em;margin-right:0.05em;",[1112,2411],{"className":2412,"style":2414},[2413],"pstrut","height:2.7em;",[1112,2416,2422],{"className":2417},[2418,2419,2420,2421],"sizing","reset-size6","size3","mtight",[1112,2423,2425],{"className":2424},[1155,2421],[1112,2426,2351],{"className":2427},[1155,2421],[1112,2429],{"className":2430,"style":1271},[1270],[1112,2432,1237],{"className":2433},[1275],[1112,2435],{"className":2436,"style":1271},[1270],[1112,2438,2440,2444,2447,2479,2482,2485,2517],{"className":2439},[1146],[1112,2441],{"className":2442,"style":2443},[1150],"height:1.0019em;vertical-align:-0.25em;",[1112,2445,1241],{"className":2446},[1289],[1112,2448,2450,2453],{"className":2449},[1155],[1112,2451,1172],{"className":2452,"style":1187},[1155,1156],[1112,2454,2456],{"className":2455},[2394],[1112,2457,2459],{"className":2458},[2398],[1112,2460,2462],{"className":2461},[2402],[1112,2463,2465],{"className":2464,"style":2384},[2406],[1112,2466,2467,2470],{"style":2409},[1112,2468],{"className":2469,"style":2414},[2413],[1112,2471,2473],{"className":2472},[2418,2419,2420,2421],[1112,2474,2476],{"className":2475},[1155,2421],[1112,2477,2351],{"className":2478},[1155,2421],[1112,2480,1246],{"className":2481},[1296],[1112,2483],{"className":2484,"style":1300},[1270],[1112,2486,2488,2491],{"className":2487},[1155],[1112,2489,1203],{"className":2490,"style":1218},[1155,1156],[1112,2492,2494],{"className":2493},[2394],[1112,2495,2497],{"className":2496},[2398],[1112,2498,2500],{"className":2499},[2402],[1112,2501,2503],{"className":2502,"style":2384},[2406],[1112,2504,2505,2508],{"style":2409},[1112,2506],{"className":2507,"style":2414},[2413],[1112,2509,2511],{"className":2510},[2418,2419,2420,2421],[1112,2512,2514],{"className":2513},[1155,2421],[1112,2515,2351],{"className":2516},[1155,2421],[1112,2518,1251],{"className":2519},[1307],"，如果  ",[1112,2522,2524,2546],{"className":2523},[1115],[1112,2525,2527],{"className":2526},[1119],[1121,2528,2529],{"xmlns":1123},[1125,2530,2531,2543],{},[1128,2532,2533,2535,2538,2541],{},[1131,2534,1172],{},[2536,2537,2351],"mtext",{},[1235,2539,2540],{},"⊆",[1131,2542,1172],{},[1135,2544,2545],{"encoding":1137},"V′⊆V",[1112,2547,2549,2571],{"className":2548,"ariaHidden":1142},[1141],[1112,2550,2552,2556,2559,2562,2565,2568],{"className":2551},[1146],[1112,2553],{"className":2554,"style":2555},[1150],"height:0.8193em;vertical-align:-0.136em;",[1112,2557,1172],{"className":2558,"style":1187},[1155,1156],[1112,2560,2351],{"className":2561},[1155],[1112,2563],{"className":2564,"style":1271},[1270],[1112,2566,2540],{"className":2567},[1275],[1112,2569],{"className":2570,"style":1271},[1270],[1112,2572,2574,2577],{"className":2573},[1146],[1112,2575],{"className":2576,"style":1151},[1150],[1112,2578,1172],{"className":2579,"style":1187},[1155,1156]," 且  ",[1112,2582,2584,2604],{"className":2583},[1115],[1112,2585,2587],{"className":2586},[1119],[1121,2588,2589],{"xmlns":1123},[1125,2590,2591,2601],{},[1128,2592,2593,2595,2597,2599],{},[1131,2594,1203],{},[2536,2596,2351],{},[1235,2598,2540],{},[1131,2600,1203],{},[1135,2602,2603],{"encoding":1137},"E′⊆E",[1112,2605,2607,2628],{"className":2606,"ariaHidden":1142},[1141],[1112,2608,2610,2613,2616,2619,2622,2625],{"className":2609},[1146],[1112,2611],{"className":2612,"style":2555},[1150],[1112,2614,1203],{"className":2615,"style":1218},[1155,1156],[1112,2617,2351],{"className":2618},[1155],[1112,2620],{"className":2621,"style":1271},[1270],[1112,2623,2540],{"className":2624},[1275],[1112,2626],{"className":2627,"style":1271},[1270],[1112,2629,2631,2634],{"className":2630},[1146],[1112,2632],{"className":2633,"style":1151},[1150],[1112,2635,1203],{"className":2636,"style":1218},[1155,1156]," ，则称 ",[1112,2639,2641,2659],{"className":2640},[1115],[1112,2642,2644],{"className":2643},[1119],[1121,2645,2646],{"xmlns":1123},[1125,2647,2648,2656],{},[1128,2649,2650],{},[2343,2651,2652,2654],{},[1131,2653,1133],{},[1235,2655,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1135,2657,2658],{"encoding":1137},"G'",[1112,2660,2662],{"className":2661,"ariaHidden":1142},[1141],[1112,2663,2665,2668],{"className":2664},[1146],[1112,2666],{"className":2667,"style":2384},[1150],[1112,2669,2671,2674],{"className":2670},[1155],[1112,2672,1133],{"className":2673},[1155,1156],[1112,2675,2677],{"className":2676},[2394],[1112,2678,2680],{"className":2679},[2398],[1112,2681,2683],{"className":2682},[2402],[1112,2684,2686],{"className":2685,"style":2384},[2406],[1112,2687,2688,2691],{"style":2409},[1112,2689],{"className":2690,"style":2414},[2413],[1112,2692,2694],{"className":2693},[2418,2419,2420,2421],[1112,2695,2697],{"className":2696},[1155,2421],[1112,2698,2351],{"className":2699},[1155,2421]," 为 ",[1112,2702,2704,2717],{"className":2703},[1115],[1112,2705,2707],{"className":2706},[1119],[1121,2708,2709],{"xmlns":1123},[1125,2710,2711,2715],{},[1128,2712,2713],{},[1131,2714,1133],{},[1135,2716,1133],{"encoding":1137},[1112,2718,2720],{"className":2719,"ariaHidden":1142},[1141],[1112,2721,2723,2726],{"className":2722},[1146],[1112,2724],{"className":2725,"style":1151},[1150],[1112,2727,1133],{"className":2728},[1155,1156]," 的子图。如下图所示，图（b）是图（a）的子图。",[1091,2731,2732],{},[1634,2733],{"alt":2039,"src":2734},"https:\u002F\u002Fpic4.zhimg.com\u002F80\u002Fv2-2c684f66ed3025d455fc56f2d25a1121_720w.webp",[2041,2736,2737,2889,2979,2982,3510],{},[2044,2738,2739,2740,2821,2822,2888],{},"无向完全图和有向完全图：对于无向图，若具有 ",[1112,2741,2743,2777],{"className":2742},[1115],[1112,2744,2746],{"className":2745},[1119],[1121,2747,2748],{"xmlns":1123},[1125,2749,2750,2774],{},[1128,2751,2752,2755,2757,2759,2762,2766,2768,2771],{},[1131,2753,2754],{},"n",[1235,2756,1241],{"stretchy":1240},[1131,2758,2754],{},[1235,2760,2761],{},"−",[2763,2764,2765],"mn",{},"1",[1235,2767,1251],{"stretchy":1240},[1131,2769,2770],{"mathvariant":2349},"\u002F",[2763,2772,2773],{},"2",[1135,2775,2776],{"encoding":1137},"n(n-1)\u002F2",[1112,2778,2780,2805],{"className":2779,"ariaHidden":1142},[1141],[1112,2781,2783,2786,2789,2792,2795,2798,2802],{"className":2782},[1146],[1112,2784],{"className":2785,"style":1285},[1150],[1112,2787,2754],{"className":2788},[1155,1156],[1112,2790,1241],{"className":2791},[1289],[1112,2793,2754],{"className":2794},[1155,1156],[1112,2796],{"className":2797,"style":1187},[1270],[1112,2799,2761],{"className":2800},[2801],"mbin",[1112,2803],{"className":2804,"style":1187},[1270],[1112,2806,2808,2811,2814,2817],{"className":2807},[1146],[1112,2809],{"className":2810,"style":1285},[1150],[1112,2812,2765],{"className":2813},[1155],[1112,2815,1251],{"className":2816},[1307],[1112,2818,2820],{"className":2819},[1155],"\u002F2"," 条边，则称为无向安全图。对于有向图，若具有 ",[1112,2823,2825,2849],{"className":2824},[1115],[1112,2826,2828],{"className":2827},[1119],[1121,2829,2830],{"xmlns":1123},[1125,2831,2832,2846],{},[1128,2833,2834,2836,2838,2840,2842,2844],{},[1131,2835,2754],{},[1235,2837,1241],{"stretchy":1240},[1131,2839,2754],{},[1235,2841,2761],{},[2763,2843,2765],{},[1235,2845,1251],{"stretchy":1240},[1135,2847,2848],{"encoding":1137},"n(n-1)",[1112,2850,2852,2876],{"className":2851,"ariaHidden":1142},[1141],[1112,2853,2855,2858,2861,2864,2867,2870,2873],{"className":2854},[1146],[1112,2856],{"className":2857,"style":1285},[1150],[1112,2859,2754],{"className":2860},[1155,1156],[1112,2862,1241],{"className":2863},[1289],[1112,2865,2754],{"className":2866},[1155,1156],[1112,2868],{"className":2869,"style":1187},[1270],[1112,2871,2761],{"className":2872},[2801],[1112,2874],{"className":2875,"style":1187},[1270],[1112,2877,2879,2882,2885],{"className":2878},[1146],[1112,2880],{"className":2881,"style":1285},[1150],[1112,2883,2765],{"className":2884},[1155],[1112,2886,1251],{"className":2887},[1307]," 条弧，则称为有向完全图。",[2044,2890,2891,2892,2978],{},"稀疏图和稠密图：边或弧很少（如  ",[1112,2893,2895,2930],{"className":2894},[1115],[1112,2896,2898],{"className":2897},[1119],[1121,2899,2900],{"xmlns":1123},[1125,2901,2902,2927],{},[1128,2903,2904,2907,2909,2911,2914,2917,2920,2922,2925],{},[1131,2905,2906],{},"e",[1235,2908,1778],{},[1131,2910,2754],{},[1131,2912,2913],{},"l",[1131,2915,2916],{},"o",[1131,2918,2919],{},"g",[2763,2921,2773],{},[2536,2923,2924],{},"⁡",[1131,2926,2754],{},[1135,2928,2929],{"encoding":1137},"e\u003Cnlog2⁡n",[1112,2931,2933,2951],{"className":2932,"ariaHidden":1142},[1141],[1112,2934,2936,2939,2942,2945,2948],{"className":2935},[1146],[1112,2937],{"className":2938,"style":1802},[1150],[1112,2940,2906],{"className":2941},[1155,1156],[1112,2943],{"className":2944,"style":1271},[1270],[1112,2946,1778],{"className":2947},[1275],[1112,2949],{"className":2950,"style":1271},[1270],[1112,2952,2954,2958,2961,2965,2968,2971,2975],{"className":2953},[1146],[1112,2955],{"className":2956,"style":2957},[1150],"height:0.8889em;vertical-align:-0.1944em;",[1112,2959,2754],{"className":2960},[1155,1156],[1112,2962,2913],{"className":2963,"style":2964},[1155,1156],"margin-right:0.0197em;",[1112,2966,2916],{"className":2967},[1155,1156],[1112,2969,2919],{"className":2970,"style":1828},[1155,1156],[1112,2972,2974],{"className":2973},[1155],"2⁡",[1112,2976,2754],{"className":2977},[1155,1156]," ​ ）的图称为稀疏图，反之称为稠密图。",[2044,2980,2981],{},"权和网：若在图的每条边上标上具有某种含义的数值，该数值称为该边上的权值。这些权值可以表示从一个顶点到另一个顶点的距离或耗费。这种带权的图通常称为网。",[2044,2983,2984,2985,3066,3067,2329,3096,3158,3159,3244,3245,2329,3273,3334,3335,3419,3420,2329,3448,3509],{},"邻接点：对于无向图 G，如果图的边  ",[1112,2986,2988,3018],{"className":2987},[1115],[1112,2989,2991],{"className":2990},[1119],[1121,2992,2993],{"xmlns":1123},[1125,2994,2995,3015],{},[1128,2996,2997,2999,3002,3004,3006,3008,3010,3013],{},[1235,2998,1241],{"stretchy":1240},[1131,3000,3001],{},"v",[1235,3003,1246],{"separator":1142},[1131,3005,3001],{},[2536,3007,2351],{},[1235,3009,1251],{"stretchy":1240},[1235,3011,3012],{},"∈",[1131,3014,1203],{},[1135,3016,3017],{"encoding":1137},"(v,v′)∈E",[1112,3019,3021,3057],{"className":3020,"ariaHidden":1142},[1141],[1112,3022,3024,3027,3030,3033,3036,3039,3042,3045,3048,3051,3054],{"className":3023},[1146],[1112,3025],{"className":3026,"style":1285},[1150],[1112,3028,1241],{"className":3029},[1289],[1112,3031,3001],{"className":3032,"style":1828},[1155,1156],[1112,3034,1246],{"className":3035},[1296],[1112,3037],{"className":3038,"style":1300},[1270],[1112,3040,3001],{"className":3041,"style":1828},[1155,1156],[1112,3043,2351],{"className":3044},[1155],[1112,3046,1251],{"className":3047},[1307],[1112,3049],{"className":3050,"style":1271},[1270],[1112,3052,3012],{"className":3053},[1275],[1112,3055],{"className":3056,"style":1271},[1270],[1112,3058,3060,3063],{"className":3059},[1146],[1112,3061],{"className":3062,"style":1151},[1150],[1112,3064,1203],{"className":3065,"style":1218},[1155,1156]," ，则称顶点 ",[1112,3068,3070,3083],{"className":3069},[1115],[1112,3071,3073],{"className":3072},[1119],[1121,3074,3075],{"xmlns":1123},[1125,3076,3077,3081],{},[1128,3078,3079],{},[1131,3080,3001],{},[1135,3082,3001],{"encoding":1137},[1112,3084,3086],{"className":3085,"ariaHidden":1142},[1141],[1112,3087,3089,3093],{"className":3088},[1146],[1112,3090],{"className":3091,"style":3092},[1150],"height:0.4306em;",[1112,3094,3001],{"className":3095,"style":1828},[1155,1156],[1112,3097,3099,3117],{"className":3098},[1115],[1112,3100,3102],{"className":3101},[1119],[1121,3103,3104],{"xmlns":1123},[1125,3105,3106,3114],{},[1128,3107,3108],{},[2343,3109,3110,3112],{},[1131,3111,3001],{},[1235,3113,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1135,3115,3116],{"encoding":1137},"v'",[1112,3118,3120],{"className":3119,"ariaHidden":1142},[1141],[1112,3121,3123,3126],{"className":3122},[1146],[1112,3124],{"className":3125,"style":2384},[1150],[1112,3127,3129,3132],{"className":3128},[1155],[1112,3130,3001],{"className":3131,"style":1828},[1155,1156],[1112,3133,3135],{"className":3134},[2394],[1112,3136,3138],{"className":3137},[2398],[1112,3139,3141],{"className":3140},[2402],[1112,3142,3144],{"className":3143,"style":2384},[2406],[1112,3145,3146,3149],{"style":2409},[1112,3147],{"className":3148,"style":2414},[2413],[1112,3150,3152],{"className":3151},[2418,2419,2420,2421],[1112,3153,3155],{"className":3154},[1155,2421],[1112,3156,2351],{"className":3157},[1155,2421]," 互为邻接点，即 v 和 v'相邻接。边",[1112,3160,3162,3188],{"className":3161},[1115],[1112,3163,3165],{"className":3164},[1119],[1121,3166,3167],{"xmlns":1123},[1125,3168,3169,3185],{},[1128,3170,3171,3173,3175,3177,3183],{},[1235,3172,1241],{"stretchy":1240},[1131,3174,3001],{},[1235,3176,1246],{"separator":1142},[2343,3178,3179,3181],{},[1131,3180,3001],{},[1235,3182,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1235,3184,1251],{"stretchy":1240},[1135,3186,3187],{"encoding":1137},"(v, v')",[1112,3189,3191],{"className":3190,"ariaHidden":1142},[1141],[1112,3192,3194,3197,3200,3203,3206,3209,3241],{"className":3193},[1146],[1112,3195],{"className":3196,"style":2443},[1150],[1112,3198,1241],{"className":3199},[1289],[1112,3201,3001],{"className":3202,"style":1828},[1155,1156],[1112,3204,1246],{"className":3205},[1296],[1112,3207],{"className":3208,"style":1300},[1270],[1112,3210,3212,3215],{"className":3211},[1155],[1112,3213,3001],{"className":3214,"style":1828},[1155,1156],[1112,3216,3218],{"className":3217},[2394],[1112,3219,3221],{"className":3220},[2398],[1112,3222,3224],{"className":3223},[2402],[1112,3225,3227],{"className":3226,"style":2384},[2406],[1112,3228,3229,3232],{"style":2409},[1112,3230],{"className":3231,"style":2414},[2413],[1112,3233,3235],{"className":3234},[2418,2419,2420,2421],[1112,3236,3238],{"className":3237},[1155,2421],[1112,3239,2351],{"className":3240},[1155,2421],[1112,3242,1251],{"className":3243},[1307],"依附于顶点 ",[1112,3246,3248,3261],{"className":3247},[1115],[1112,3249,3251],{"className":3250},[1119],[1121,3252,3253],{"xmlns":1123},[1125,3254,3255,3259],{},[1128,3256,3257],{},[1131,3258,3001],{},[1135,3260,3001],{"encoding":1137},[1112,3262,3264],{"className":3263,"ariaHidden":1142},[1141],[1112,3265,3267,3270],{"className":3266},[1146],[1112,3268],{"className":3269,"style":3092},[1150],[1112,3271,3001],{"className":3272,"style":1828},[1155,1156],[1112,3274,3276,3293],{"className":3275},[1115],[1112,3277,3279],{"className":3278},[1119],[1121,3280,3281],{"xmlns":1123},[1125,3282,3283,3291],{},[1128,3284,3285],{},[2343,3286,3287,3289],{},[1131,3288,3001],{},[1235,3290,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1135,3292,3116],{"encoding":1137},[1112,3294,3296],{"className":3295,"ariaHidden":1142},[1141],[1112,3297,3299,3302],{"className":3298},[1146],[1112,3300],{"className":3301,"style":2384},[1150],[1112,3303,3305,3308],{"className":3304},[1155],[1112,3306,3001],{"className":3307,"style":1828},[1155,1156],[1112,3309,3311],{"className":3310},[2394],[1112,3312,3314],{"className":3313},[2398],[1112,3315,3317],{"className":3316},[2402],[1112,3318,3320],{"className":3319,"style":2384},[2406],[1112,3321,3322,3325],{"style":2409},[1112,3323],{"className":3324,"style":2414},[2413],[1112,3326,3328],{"className":3327},[2418,2419,2420,2421],[1112,3329,3331],{"className":3330},[1155,2421],[1112,3332,2351],{"className":3333},[1155,2421],"，或者说边",[1112,3336,3338,3363],{"className":3337},[1115],[1112,3339,3341],{"className":3340},[1119],[1121,3342,3343],{"xmlns":1123},[1125,3344,3345,3361],{},[1128,3346,3347,3349,3351,3353,3359],{},[1235,3348,1241],{"stretchy":1240},[1131,3350,3001],{},[1235,3352,1246],{"separator":1142},[2343,3354,3355,3357],{},[1131,3356,3001],{},[1235,3358,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1235,3360,1251],{"stretchy":1240},[1135,3362,3187],{"encoding":1137},[1112,3364,3366],{"className":3365,"ariaHidden":1142},[1141],[1112,3367,3369,3372,3375,3378,3381,3384,3416],{"className":3368},[1146],[1112,3370],{"className":3371,"style":2443},[1150],[1112,3373,1241],{"className":3374},[1289],[1112,3376,3001],{"className":3377,"style":1828},[1155,1156],[1112,3379,1246],{"className":3380},[1296],[1112,3382],{"className":3383,"style":1300},[1270],[1112,3385,3387,3390],{"className":3386},[1155],[1112,3388,3001],{"className":3389,"style":1828},[1155,1156],[1112,3391,3393],{"className":3392},[2394],[1112,3394,3396],{"className":3395},[2398],[1112,3397,3399],{"className":3398},[2402],[1112,3400,3402],{"className":3401,"style":2384},[2406],[1112,3403,3404,3407],{"style":2409},[1112,3405],{"className":3406,"style":2414},[2413],[1112,3408,3410],{"className":3409},[2418,2419,2420,2421],[1112,3411,3413],{"className":3412},[1155,2421],[1112,3414,2351],{"className":3415},[1155,2421],[1112,3417,1251],{"className":3418},[1307],"与顶点 ",[1112,3421,3423,3436],{"className":3422},[1115],[1112,3424,3426],{"className":3425},[1119],[1121,3427,3428],{"xmlns":1123},[1125,3429,3430,3434],{},[1128,3431,3432],{},[1131,3433,3001],{},[1135,3435,3001],{"encoding":1137},[1112,3437,3439],{"className":3438,"ariaHidden":1142},[1141],[1112,3440,3442,3445],{"className":3441},[1146],[1112,3443],{"className":3444,"style":3092},[1150],[1112,3446,3001],{"className":3447,"style":1828},[1155,1156],[1112,3449,3451,3468],{"className":3450},[1115],[1112,3452,3454],{"className":3453},[1119],[1121,3455,3456],{"xmlns":1123},[1125,3457,3458,3466],{},[1128,3459,3460],{},[2343,3461,3462,3464],{},[1131,3463,3001],{},[1235,3465,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1135,3467,3116],{"encoding":1137},[1112,3469,3471],{"className":3470,"ariaHidden":1142},[1141],[1112,3472,3474,3477],{"className":3473},[1146],[1112,3475],{"className":3476,"style":2384},[1150],[1112,3478,3480,3483],{"className":3479},[1155],[1112,3481,3001],{"className":3482,"style":1828},[1155,1156],[1112,3484,3486],{"className":3485},[2394],[1112,3487,3489],{"className":3488},[2398],[1112,3490,3492],{"className":3491},[2402],[1112,3493,3495],{"className":3494,"style":2384},[2406],[1112,3496,3497,3500],{"style":2409},[1112,3498],{"className":3499,"style":2414},[2413],[1112,3501,3503],{"className":3502},[2418,2419,2420,2421],[1112,3504,3506],{"className":3505},[1155,2421],[1112,3507,2351],{"className":3508},[1155,2421]," 相关联。",[2044,3511,3512,3513,3541,3542,3570,3571,3624,3625,3703,3704,3707,3708,3759,3760,3810,3811,3945,3946,3948,3949,4018,4019,4144,4145,4267,4268],{},"度、入度和出度：顶点 ",[1112,3514,3516,3529],{"className":3515},[1115],[1112,3517,3519],{"className":3518},[1119],[1121,3520,3521],{"xmlns":1123},[1125,3522,3523,3527],{},[1128,3524,3525],{},[1131,3526,3001],{},[1135,3528,3001],{"encoding":1137},[1112,3530,3532],{"className":3531,"ariaHidden":1142},[1141],[1112,3533,3535,3538],{"className":3534},[1146],[1112,3536],{"className":3537,"style":3092},[1150],[1112,3539,3001],{"className":3540,"style":1828},[1155,1156]," 的度是指和 ",[1112,3543,3545,3558],{"className":3544},[1115],[1112,3546,3548],{"className":3547},[1119],[1121,3549,3550],{"xmlns":1123},[1125,3551,3552,3556],{},[1128,3553,3554],{},[1131,3555,3001],{},[1135,3557,3001],{"encoding":1137},[1112,3559,3561],{"className":3560,"ariaHidden":1142},[1141],[1112,3562,3564,3567],{"className":3563},[1146],[1112,3565],{"className":3566,"style":3092},[1150],[1112,3568,3001],{"className":3569,"style":1828},[1155,1156]," 相关联的边的数目，记为 ",[1112,3572,3574,3598],{"className":3573},[1115],[1112,3575,3577],{"className":3576},[1119],[1121,3578,3579],{"xmlns":1123},[1125,3580,3581,3595],{},[1128,3582,3583,3586,3589,3591,3593],{},[1131,3584,3585],{},"T",[1131,3587,3588],{},"D",[1235,3590,1241],{"stretchy":1240},[1131,3592,3001],{},[1235,3594,1251],{"stretchy":1240},[1135,3596,3597],{"encoding":1137},"TD(v)",[1112,3599,3601],{"className":3600,"ariaHidden":1142},[1141],[1112,3602,3604,3607,3611,3615,3618,3621],{"className":3603},[1146],[1112,3605],{"className":3606,"style":1285},[1150],[1112,3608,3585],{"className":3609,"style":3610},[1155,1156],"margin-right:0.1389em;",[1112,3612,3588],{"className":3613,"style":3614},[1155,1156],"margin-right:0.0278em;",[1112,3616,1241],{"className":3617},[1289],[1112,3619,3001],{"className":3620,"style":1828},[1155,1156],[1112,3622,1251],{"className":3623},[1307],"。例如，下图（b）中的顶点 ",[1112,3626,3628,3647],{"className":3627},[1115],[1112,3629,3631],{"className":3630},[1119],[1121,3632,3633],{"xmlns":1123},[1125,3634,3635,3644],{},[1128,3636,3637],{},[3638,3639,3640,3642],"msub",{},[1131,3641,3001],{},[2763,3643,2765],{},[1135,3645,3646],{"encoding":1137},"v_1",[1112,3648,3650],{"className":3649,"ariaHidden":1142},[1141],[1112,3651,3653,3657],{"className":3652},[1146],[1112,3654],{"className":3655,"style":3656},[1150],"height:0.5806em;vertical-align:-0.15em;",[1112,3658,3660,3663],{"className":3659},[1155],[1112,3661,3001],{"className":3662,"style":1828},[1155,1156],[1112,3664,3666],{"className":3665},[2394],[1112,3667,3670,3694],{"className":3668},[2398,3669],"vlist-t2",[1112,3671,3673,3689],{"className":3672},[2402],[1112,3674,3677],{"className":3675,"style":3676},[2406],"height:0.3011em;",[1112,3678,3680,3683],{"style":3679},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[1112,3681],{"className":3682,"style":2414},[2413],[1112,3684,3686],{"className":3685},[2418,2419,2420,2421],[1112,3687,2765],{"className":3688},[1155,2421],[1112,3690,3693],{"className":3691},[3692],"vlist-s","​",[1112,3695,3697],{"className":3696},[2402],[1112,3698,3701],{"className":3699,"style":3700},[2406],"height:0.15em;",[1112,3702],{},"​ 的度是 2。",[3705,3706],"br",{},"对于有向图，顶点 v 的度分为入度和出度。\n入度是以顶点 v 为头的弧的数目，记为 ",[1112,3709,3711,3734],{"className":3710},[1115],[1112,3712,3714],{"className":3713},[1119],[1121,3715,3716],{"xmlns":1123},[1125,3717,3718,3731],{},[1128,3719,3720,3723,3725,3727,3729],{},[1131,3721,3722],{},"I",[1131,3724,3588],{},[1235,3726,1241],{"stretchy":1240},[1131,3728,3001],{},[1235,3730,1251],{"stretchy":1240},[1135,3732,3733],{"encoding":1137},"ID(v)",[1112,3735,3737],{"className":3736,"ariaHidden":1142},[1141],[1112,3738,3740,3743,3747,3750,3753,3756],{"className":3739},[1146],[1112,3741],{"className":3742,"style":1285},[1150],[1112,3744,3722],{"className":3745,"style":3746},[1155,1156],"margin-right:0.0785em;",[1112,3748,3588],{"className":3749,"style":3614},[1155,1156],[1112,3751,1241],{"className":3752},[1289],[1112,3754,3001],{"className":3755,"style":1828},[1155,1156],[1112,3757,1251],{"className":3758},[1307],"；\n出度是以顶点 v 为尾的弧的数目，记为 ",[1112,3761,3763,3786],{"className":3762},[1115],[1112,3764,3766],{"className":3765},[1119],[1121,3767,3768],{"xmlns":1123},[1125,3769,3770,3783],{},[1128,3771,3772,3775,3777,3779,3781],{},[1131,3773,3774],{},"O",[1131,3776,3588],{},[1235,3778,1241],{"stretchy":1240},[1131,3780,3001],{},[1235,3782,1251],{"stretchy":1240},[1135,3784,3785],{"encoding":1137},"OD(v)",[1112,3787,3789],{"className":3788,"ariaHidden":1142},[1141],[1112,3790,3792,3795,3798,3801,3804,3807],{"className":3791},[1146],[1112,3793],{"className":3794,"style":1285},[1150],[1112,3796,3774],{"className":3797,"style":3614},[1155,1156],[1112,3799,3588],{"className":3800,"style":3614},[1155,1156],[1112,3802,1241],{"className":3803},[1289],[1112,3805,3001],{"className":3806,"style":1828},[1155,1156],[1112,3808,1251],{"className":3809},[1307]," 。\n顶点 v 的度为 ",[1112,3812,3814,3861],{"className":3813},[1115],[1112,3815,3817],{"className":3816},[1119],[1121,3818,3819],{"xmlns":1123},[1125,3820,3821,3858],{},[1128,3822,3823,3825,3827,3829,3831,3833,3835,3837,3839,3841,3843,3845,3848,3850,3852,3854,3856],{},[1131,3824,3585],{},[1131,3826,3588],{},[1235,3828,1241],{"stretchy":1240},[1131,3830,3001],{},[1235,3832,1251],{"stretchy":1240},[1235,3834,1237],{},[1131,3836,3722],{},[1131,3838,3588],{},[1235,3840,1241],{"stretchy":1240},[1131,3842,3001],{},[1235,3844,1251],{"stretchy":1240},[1235,3846,3847],{},"+",[1131,3849,3774],{},[1131,3851,3588],{},[1235,3853,1241],{"stretchy":1240},[1131,3855,3001],{},[1235,3857,1251],{"stretchy":1240},[1135,3859,3860],{"encoding":1137},"TD(v)=ID(v)+OD(v)",[1112,3862,3864,3894,3924],{"className":3863,"ariaHidden":1142},[1141],[1112,3865,3867,3870,3873,3876,3879,3882,3885,3888,3891],{"className":3866},[1146],[1112,3868],{"className":3869,"style":1285},[1150],[1112,3871,3585],{"className":3872,"style":3610},[1155,1156],[1112,3874,3588],{"className":3875,"style":3614},[1155,1156],[1112,3877,1241],{"className":3878},[1289],[1112,3880,3001],{"className":3881,"style":1828},[1155,1156],[1112,3883,1251],{"className":3884},[1307],[1112,3886],{"className":3887,"style":1271},[1270],[1112,3889,1237],{"className":3890},[1275],[1112,3892],{"className":3893,"style":1271},[1270],[1112,3895,3897,3900,3903,3906,3909,3912,3915,3918,3921],{"className":3896},[1146],[1112,3898],{"className":3899,"style":1285},[1150],[1112,3901,3722],{"className":3902,"style":3746},[1155,1156],[1112,3904,3588],{"className":3905,"style":3614},[1155,1156],[1112,3907,1241],{"className":3908},[1289],[1112,3910,3001],{"className":3911,"style":1828},[1155,1156],[1112,3913,1251],{"className":3914},[1307],[1112,3916],{"className":3917,"style":1187},[1270],[1112,3919,3847],{"className":3920},[2801],[1112,3922],{"className":3923,"style":1187},[1270],[1112,3925,3927,3930,3933,3936,3939,3942],{"className":3926},[1146],[1112,3928],{"className":3929,"style":1285},[1150],[1112,3931,3774],{"className":3932,"style":3614},[1155,1156],[1112,3934,3588],{"className":3935,"style":3614},[1155,1156],[1112,3937,1241],{"className":3938},[1289],[1112,3940,3001],{"className":3941,"style":1828},[1155,1156],[1112,3943,1251],{"className":3944},[1307],"。",[3705,3947],{},"例如，下图（a）中的顶点  ",[1112,3950,3952,3969],{"className":3951},[1115],[1112,3953,3955],{"className":3954},[1119],[1121,3956,3957],{"xmlns":1123},[1125,3958,3959,3967],{},[1128,3960,3961],{},[3638,3962,3963,3965],{},[1131,3964,3001],{},[2763,3966,2765],{},[1135,3968,3646],{"encoding":1137},[1112,3970,3972],{"className":3971,"ariaHidden":1142},[1141],[1112,3973,3975,3978],{"className":3974},[1146],[1112,3976],{"className":3977,"style":3656},[1150],[1112,3979,3981,3984],{"className":3980},[1155],[1112,3982,3001],{"className":3983,"style":1828},[1155,1156],[1112,3985,3987],{"className":3986},[2394],[1112,3988,3990,4010],{"className":3989},[2398,3669],[1112,3991,3993,4007],{"className":3992},[2402],[1112,3994,3996],{"className":3995,"style":3676},[2406],[1112,3997,3998,4001],{"style":3679},[1112,3999],{"className":4000,"style":2414},[2413],[1112,4002,4004],{"className":4003},[2418,2419,2420,2421],[1112,4005,2765],{"className":4006},[1155,2421],[1112,4008,3693],{"className":4009},[3692],[1112,4011,4013],{"className":4012},[2402],[1112,4014,4016],{"className":4015,"style":3700},[2406],[1112,4017],{}," ​ 的入度 ",[1112,4020,4022,4058],{"className":4021},[1115],[1112,4023,4025],{"className":4024},[1119],[1121,4026,4027],{"xmlns":1123},[1125,4028,4029,4055],{},[1128,4030,4031,4033,4035,4037,4040,4046,4049,4051,4053],{},[1131,4032,3722],{},[1131,4034,3588],{},[1235,4036,1241],{"stretchy":1240},[2536,4038,4039],{},"​ ",[3638,4041,4042,4044],{},[1131,4043,3001],{},[2763,4045,2765],{},[2536,4047,4048],{}," ",[1235,4050,1251],{"stretchy":1240},[1235,4052,1237],{},[2763,4054,2765],{},[1135,4056,4057],{"encoding":1137},"ID(​ v_1 )=1",[1112,4059,4061,4134],{"className":4060,"ariaHidden":1142},[1141],[1112,4062,4064,4067,4070,4073,4076,4079,4119,4122,4125,4128,4131],{"className":4063},[1146],[1112,4065],{"className":4066,"style":1285},[1150],[1112,4068,3722],{"className":4069,"style":3746},[1155,1156],[1112,4071,3588],{"className":4072,"style":3614},[1155,1156],[1112,4074,1241],{"className":4075},[1289],[1112,4077,4039],{"className":4078},[1155],[1112,4080,4082,4085],{"className":4081},[1155],[1112,4083,3001],{"className":4084,"style":1828},[1155,1156],[1112,4086,4088],{"className":4087},[2394],[1112,4089,4091,4111],{"className":4090},[2398,3669],[1112,4092,4094,4108],{"className":4093},[2402],[1112,4095,4097],{"className":4096,"style":3676},[2406],[1112,4098,4099,4102],{"style":3679},[1112,4100],{"className":4101,"style":2414},[2413],[1112,4103,4105],{"className":4104},[2418,2419,2420,2421],[1112,4106,2765],{"className":4107},[1155,2421],[1112,4109,3693],{"className":4110},[3692],[1112,4112,4114],{"className":4113},[2402],[1112,4115,4117],{"className":4116,"style":3700},[2406],[1112,4118],{},[1112,4120,4048],{"className":4121},[1155],[1112,4123,1251],{"className":4124},[1307],[1112,4126],{"className":4127,"style":1271},[1270],[1112,4129,1237],{"className":4130},[1275],[1112,4132],{"className":4133,"style":1271},[1270],[1112,4135,4137,4141],{"className":4136},[1146],[1112,4138],{"className":4139,"style":4140},[1150],"height:0.6444em;",[1112,4142,2765],{"className":4143},[1155],"，出度 ",[1112,4146,4148,4182],{"className":4147},[1115],[1112,4149,4151],{"className":4150},[1119],[1121,4152,4153],{"xmlns":1123},[1125,4154,4155,4179],{},[1128,4156,4157,4159,4161,4163,4165,4171,4173,4175,4177],{},[1131,4158,3774],{},[1131,4160,3588],{},[1235,4162,1241],{"stretchy":1240},[2536,4164,4039],{},[3638,4166,4167,4169],{},[1131,4168,3001],{},[2763,4170,2765],{},[2536,4172,4048],{},[1235,4174,1251],{"stretchy":1240},[1235,4176,1237],{},[2763,4178,2773],{},[1135,4180,4181],{"encoding":1137},"OD(​ v_1 )=2",[1112,4183,4185,4258],{"className":4184,"ariaHidden":1142},[1141],[1112,4186,4188,4191,4194,4197,4200,4203,4243,4246,4249,4252,4255],{"className":4187},[1146],[1112,4189],{"className":4190,"style":1285},[1150],[1112,4192,3774],{"className":4193,"style":3614},[1155,1156],[1112,4195,3588],{"className":4196,"style":3614},[1155,1156],[1112,4198,1241],{"className":4199},[1289],[1112,4201,4039],{"className":4202},[1155],[1112,4204,4206,4209],{"className":4205},[1155],[1112,4207,3001],{"className":4208,"style":1828},[1155,1156],[1112,4210,4212],{"className":4211},[2394],[1112,4213,4215,4235],{"className":4214},[2398,3669],[1112,4216,4218,4232],{"className":4217},[2402],[1112,4219,4221],{"className":4220,"style":3676},[2406],[1112,4222,4223,4226],{"style":3679},[1112,4224],{"className":4225,"style":2414},[2413],[1112,4227,4229],{"className":4228},[2418,2419,2420,2421],[1112,4230,2765],{"className":4231},[1155,2421],[1112,4233,3693],{"className":4234},[3692],[1112,4236,4238],{"className":4237},[2402],[1112,4239,4241],{"className":4240,"style":3700},[2406],[1112,4242],{},[1112,4244,4048],{"className":4245},[1155],[1112,4247,1251],{"className":4248},[1307],[1112,4250],{"className":4251,"style":1271},[1270],[1112,4253,1237],{"className":4254},[1275],[1112,4256],{"className":4257,"style":1271},[1270],[1112,4259,4261,4264],{"className":4260},[1146],[1112,4262],{"className":4263,"style":4140},[1150],[1112,4265,2773],{"className":4266},[1155],"，度 ",[1112,4269,4271,4347],{"className":4270},[1115],[1112,4272,4274],{"className":4273},[1119],[1121,4275,4276],{"xmlns":1123},[1125,4277,4278,4344],{},[1128,4279,4280,4282,4284,4286,4288,4294,4296,4298,4300,4302,4304,4306,4308,4314,4317,4319,4321,4323,4325,4327,4329,4335,4337,4339,4341],{},[1131,4281,3585],{},[1131,4283,3588],{},[1235,4285,1241],{"stretchy":1240},[2536,4287,4039],{},[3638,4289,4290,4292],{},[1131,4291,3001],{},[2763,4293,2765],{},[2536,4295,4048],{},[1235,4297,1251],{"stretchy":1240},[1235,4299,1237],{},[1131,4301,3722],{},[1131,4303,3588],{},[1235,4305,1241],{"stretchy":1240},[2536,4307,4048],{},[3638,4309,4310,4312],{},[1131,4311,3001],{},[2763,4313,2765],{},[2536,4315,4316],{}," ​",[1235,4318,1251],{"stretchy":1240},[1235,4320,3847],{},[1131,4322,3774],{},[1131,4324,3588],{},[1235,4326,1241],{"stretchy":1240},[2536,4328,4048],{},[3638,4330,4331,4333],{},[1131,4332,3001],{},[2763,4334,2765],{},[2536,4336,4316],{},[1235,4338,1251],{"stretchy":1240},[1235,4340,1237],{},[2763,4342,4343],{},"3",[1135,4345,4346],{"encoding":1137},"TD(​ v_1 )=ID( v_1 ​)+OD( v_1 ​)=3",[1112,4348,4350,4423,4496,4569],{"className":4349,"ariaHidden":1142},[1141],[1112,4351,4353,4356,4359,4362,4365,4368,4408,4411,4414,4417,4420],{"className":4352},[1146],[1112,4354],{"className":4355,"style":1285},[1150],[1112,4357,3585],{"className":4358,"style":3610},[1155,1156],[1112,4360,3588],{"className":4361,"style":3614},[1155,1156],[1112,4363,1241],{"className":4364},[1289],[1112,4366,4039],{"className":4367},[1155],[1112,4369,4371,4374],{"className":4370},[1155],[1112,4372,3001],{"className":4373,"style":1828},[1155,1156],[1112,4375,4377],{"className":4376},[2394],[1112,4378,4380,4400],{"className":4379},[2398,3669],[1112,4381,4383,4397],{"className":4382},[2402],[1112,4384,4386],{"className":4385,"style":3676},[2406],[1112,4387,4388,4391],{"style":3679},[1112,4389],{"className":4390,"style":2414},[2413],[1112,4392,4394],{"className":4393},[2418,2419,2420,2421],[1112,4395,2765],{"className":4396},[1155,2421],[1112,4398,3693],{"className":4399},[3692],[1112,4401,4403],{"className":4402},[2402],[1112,4404,4406],{"className":4405,"style":3700},[2406],[1112,4407],{},[1112,4409,4048],{"className":4410},[1155],[1112,4412,1251],{"className":4413},[1307],[1112,4415],{"className":4416,"style":1271},[1270],[1112,4418,1237],{"className":4419},[1275],[1112,4421],{"className":4422,"style":1271},[1270],[1112,4424,4426,4429,4432,4435,4438,4441,4481,4484,4487,4490,4493],{"className":4425},[1146],[1112,4427],{"className":4428,"style":1285},[1150],[1112,4430,3722],{"className":4431,"style":3746},[1155,1156],[1112,4433,3588],{"className":4434,"style":3614},[1155,1156],[1112,4436,1241],{"className":4437},[1289],[1112,4439,4048],{"className":4440},[1155],[1112,4442,4444,4447],{"className":4443},[1155],[1112,4445,3001],{"className":4446,"style":1828},[1155,1156],[1112,4448,4450],{"className":4449},[2394],[1112,4451,4453,4473],{"className":4452},[2398,3669],[1112,4454,4456,4470],{"className":4455},[2402],[1112,4457,4459],{"className":4458,"style":3676},[2406],[1112,4460,4461,4464],{"style":3679},[1112,4462],{"className":4463,"style":2414},[2413],[1112,4465,4467],{"className":4466},[2418,2419,2420,2421],[1112,4468,2765],{"className":4469},[1155,2421],[1112,4471,3693],{"className":4472},[3692],[1112,4474,4476],{"className":4475},[2402],[1112,4477,4479],{"className":4478,"style":3700},[2406],[1112,4480],{},[1112,4482,4316],{"className":4483},[1155],[1112,4485,1251],{"className":4486},[1307],[1112,4488],{"className":4489,"style":1187},[1270],[1112,4491,3847],{"className":4492},[2801],[1112,4494],{"className":4495,"style":1187},[1270],[1112,4497,4499,4502,4505,4508,4511,4514,4554,4557,4560,4563,4566],{"className":4498},[1146],[1112,4500],{"className":4501,"style":1285},[1150],[1112,4503,3774],{"className":4504,"style":3614},[1155,1156],[1112,4506,3588],{"className":4507,"style":3614},[1155,1156],[1112,4509,1241],{"className":4510},[1289],[1112,4512,4048],{"className":4513},[1155],[1112,4515,4517,4520],{"className":4516},[1155],[1112,4518,3001],{"className":4519,"style":1828},[1155,1156],[1112,4521,4523],{"className":4522},[2394],[1112,4524,4526,4546],{"className":4525},[2398,3669],[1112,4527,4529,4543],{"className":4528},[2402],[1112,4530,4532],{"className":4531,"style":3676},[2406],[1112,4533,4534,4537],{"style":3679},[1112,4535],{"className":4536,"style":2414},[2413],[1112,4538,4540],{"className":4539},[2418,2419,2420,2421],[1112,4541,2765],{"className":4542},[1155,2421],[1112,4544,3693],{"className":4545},[3692],[1112,4547,4549],{"className":4548},[2402],[1112,4550,4552],{"className":4551,"style":3700},[2406],[1112,4553],{},[1112,4555,4316],{"className":4556},[1155],[1112,4558,1251],{"className":4559},[1307],[1112,4561],{"className":4562,"style":1271},[1270],[1112,4564,1237],{"className":4565},[1275],[1112,4567],{"className":4568,"style":1271},[1270],[1112,4570,4572,4575],{"className":4571},[1146],[1112,4573],{"className":4574,"style":4140},[1150],[1112,4576,4343],{"className":4577},[1155],[1091,4579,4580],{},[1634,4581],{"alt":2039,"src":1637},[2041,4583,4584,5110,5326,5329,5332],{},[2044,4585,4586,4587,4615,4616,4644,4645,4706,4707,4898,4899,5033,5034,5109],{},"路径和路径长度：在无向图 ",[1112,4588,4590,4603],{"className":4589},[1115],[1112,4591,4593],{"className":4592},[1119],[1121,4594,4595],{"xmlns":1123},[1125,4596,4597,4601],{},[1128,4598,4599],{},[1131,4600,1133],{},[1135,4602,1133],{"encoding":1137},[1112,4604,4606],{"className":4605,"ariaHidden":1142},[1141],[1112,4607,4609,4612],{"className":4608},[1146],[1112,4610],{"className":4611,"style":1151},[1150],[1112,4613,1133],{"className":4614},[1155,1156]," 中，从顶点 ",[1112,4617,4619,4632],{"className":4618},[1115],[1112,4620,4622],{"className":4621},[1119],[1121,4623,4624],{"xmlns":1123},[1125,4625,4626,4630],{},[1128,4627,4628],{},[1131,4629,3001],{},[1135,4631,3001],{"encoding":1137},[1112,4633,4635],{"className":4634,"ariaHidden":1142},[1141],[1112,4636,4638,4641],{"className":4637},[1146],[1112,4639],{"className":4640,"style":3092},[1150],[1112,4642,3001],{"className":4643,"style":1828},[1155,1156]," 到顶点 ",[1112,4646,4648,4665],{"className":4647},[1115],[1112,4649,4651],{"className":4650},[1119],[1121,4652,4653],{"xmlns":1123},[1125,4654,4655,4663],{},[1128,4656,4657],{},[2343,4658,4659,4661],{},[1131,4660,3001],{},[1235,4662,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1135,4664,3116],{"encoding":1137},[1112,4666,4668],{"className":4667,"ariaHidden":1142},[1141],[1112,4669,4671,4674],{"className":4670},[1146],[1112,4672],{"className":4673,"style":2384},[1150],[1112,4675,4677,4680],{"className":4676},[1155],[1112,4678,3001],{"className":4679,"style":1828},[1155,1156],[1112,4681,4683],{"className":4682},[2394],[1112,4684,4686],{"className":4685},[2398],[1112,4687,4689],{"className":4688},[2402],[1112,4690,4692],{"className":4691,"style":2384},[2406],[1112,4693,4694,4697],{"style":2409},[1112,4695],{"className":4696,"style":2414},[2413],[1112,4698,4700],{"className":4699},[2418,2419,2420,2421],[1112,4701,4703],{"className":4702},[1155,2421],[1112,4704,2351],{"className":4705},[1155,2421]," 的路径是一个顶点序列 ​ ",[1112,4708,4710,4776],{"className":4709},[1115],[1112,4711,4713],{"className":4712},[1119],[1121,4714,4715],{"xmlns":1123},[1125,4716,4717,4773],{},[1128,4718,4719,4721,4723,4725,4727,4730,4732,4735,4737,4739,4741,4743,4745,4747,4750,4752,4754,4756,4758,4760,4762,4765,4767,4769,4771],{},[1235,4720,1241],{"stretchy":1240},[1131,4722,3001],{},[1235,4724,1237],{},[1131,4726,3001],{},[1131,4728,4729],{},"i",[1235,4731,1246],{"separator":1142},[2763,4733,4734],{},"0",[1235,4736,1246],{"separator":1142},[1131,4738,3001],{},[1131,4740,4729],{},[1235,4742,1246],{"separator":1142},[2763,4744,2765],{},[1235,4746,1246],{"separator":1142},[1131,4748,4749],{"mathvariant":2349},".",[1131,4751,4749],{"mathvariant":2349},[1131,4753,4749],{"mathvariant":2349},[1235,4755,1246],{"separator":1142},[1131,4757,3001],{},[1131,4759,4729],{},[1235,4761,1246],{"separator":1142},[1131,4763,4764],{},"m",[1235,4766,1237],{},[1131,4768,3001],{},[2536,4770,2351],{},[1235,4772,1251],{"stretchy":1240},[1135,4774,4775],{"encoding":1137},"(v=vi,0,vi,1,...,vi,m=v′)",[1112,4777,4779,4800,4883],{"className":4778,"ariaHidden":1142},[1141],[1112,4780,4782,4785,4788,4791,4794,4797],{"className":4781},[1146],[1112,4783],{"className":4784,"style":1285},[1150],[1112,4786,1241],{"className":4787},[1289],[1112,4789,3001],{"className":4790,"style":1828},[1155,1156],[1112,4792],{"className":4793,"style":1271},[1270],[1112,4795,1237],{"className":4796},[1275],[1112,4798],{"className":4799,"style":1271},[1270],[1112,4801,4803,4807,4810,4813,4816,4819,4822,4825,4828,4831,4834,4837,4840,4843,4846,4849,4853,4856,4859,4862,4865,4868,4871,4874,4877,4880],{"className":4802},[1146],[1112,4804],{"className":4805,"style":4806},[1150],"height:0.854em;vertical-align:-0.1944em;",[1112,4808,3001],{"className":4809,"style":1828},[1155,1156],[1112,4811,4729],{"className":4812},[1155,1156],[1112,4814,1246],{"className":4815},[1296],[1112,4817],{"className":4818,"style":1300},[1270],[1112,4820,4734],{"className":4821},[1155],[1112,4823,1246],{"className":4824},[1296],[1112,4826],{"className":4827,"style":1300},[1270],[1112,4829,3001],{"className":4830,"style":1828},[1155,1156],[1112,4832,4729],{"className":4833},[1155,1156],[1112,4835,1246],{"className":4836},[1296],[1112,4838],{"className":4839,"style":1300},[1270],[1112,4841,2765],{"className":4842},[1155],[1112,4844,1246],{"className":4845},[1296],[1112,4847],{"className":4848,"style":1300},[1270],[1112,4850,4852],{"className":4851},[1155],"...",[1112,4854,1246],{"className":4855},[1296],[1112,4857],{"className":4858,"style":1300},[1270],[1112,4860,3001],{"className":4861,"style":1828},[1155,1156],[1112,4863,4729],{"className":4864},[1155,1156],[1112,4866,1246],{"className":4867},[1296],[1112,4869],{"className":4870,"style":1300},[1270],[1112,4872,4764],{"className":4873},[1155,1156],[1112,4875],{"className":4876,"style":1271},[1270],[1112,4878,1237],{"className":4879},[1275],[1112,4881],{"className":4882,"style":1271},[1270],[1112,4884,4886,4889,4892,4895],{"className":4885},[1146],[1112,4887],{"className":4888,"style":1285},[1150],[1112,4890,3001],{"className":4891,"style":1828},[1155,1156],[1112,4893,2351],{"className":4894},[1155],[1112,4896,1251],{"className":4897},[1307]," ，其中 ​ ",[1112,4900,4902,4945],{"className":4901},[1115],[1112,4903,4905],{"className":4904},[1119],[1121,4906,4907],{"xmlns":1123},[1125,4908,4909,4942],{},[1128,4910,4911,4913,4915,4917,4919,4922,4924,4926,4928,4930,4932,4934,4936,4938,4940],{},[1235,4912,1241],{"stretchy":1240},[1131,4914,3001],{},[1131,4916,4729],{},[1235,4918,1246],{"separator":1142},[1131,4920,4921],{},"j",[1235,4923,2761],{},[2763,4925,2765],{},[1235,4927,1246],{"separator":1142},[1131,4929,3001],{},[1131,4931,4729],{},[1235,4933,1246],{"separator":1142},[1131,4935,4921],{},[1235,4937,1251],{"stretchy":1240},[1235,4939,3012],{},[1131,4941,1203],{},[1135,4943,4944],{"encoding":1137},"(vi,j−1,vi,j)∈E",[1112,4946,4948,4982,5024],{"className":4947,"ariaHidden":1142},[1141],[1112,4949,4951,4954,4957,4960,4963,4966,4969,4973,4976,4979],{"className":4950},[1146],[1112,4952],{"className":4953,"style":1285},[1150],[1112,4955,1241],{"className":4956},[1289],[1112,4958,3001],{"className":4959,"style":1828},[1155,1156],[1112,4961,4729],{"className":4962},[1155,1156],[1112,4964,1246],{"className":4965},[1296],[1112,4967],{"className":4968,"style":1300},[1270],[1112,4970,4921],{"className":4971,"style":4972},[1155,1156],"margin-right:0.0572em;",[1112,4974],{"className":4975,"style":1187},[1270],[1112,4977,2761],{"className":4978},[2801],[1112,4980],{"className":4981,"style":1187},[1270],[1112,4983,4985,4988,4991,4994,4997,5000,5003,5006,5009,5012,5015,5018,5021],{"className":4984},[1146],[1112,4986],{"className":4987,"style":1285},[1150],[1112,4989,2765],{"className":4990},[1155],[1112,4992,1246],{"className":4993},[1296],[1112,4995],{"className":4996,"style":1300},[1270],[1112,4998,3001],{"className":4999,"style":1828},[1155,1156],[1112,5001,4729],{"className":5002},[1155,1156],[1112,5004,1246],{"className":5005},[1296],[1112,5007],{"className":5008,"style":1300},[1270],[1112,5010,4921],{"className":5011,"style":4972},[1155,1156],[1112,5013,1251],{"className":5014},[1307],[1112,5016],{"className":5017,"style":1271},[1270],[1112,5019,3012],{"className":5020},[1275],[1112,5022],{"className":5023,"style":1271},[1270],[1112,5025,5027,5030],{"className":5026},[1146],[1112,5028],{"className":5029,"style":1151},[1150],[1112,5031,1203],{"className":5032,"style":1218},[1155,1156]," ， ",[1112,5035,5037,5060],{"className":5036},[1115],[1112,5038,5040],{"className":5039},[1119],[1121,5041,5042],{"xmlns":1123},[1125,5043,5044,5057],{},[1128,5045,5046,5048,5051,5053,5055],{},[2763,5047,2765],{},[1235,5049,5050],{},"≤",[1131,5052,4921],{},[1235,5054,5050],{},[1131,5056,4764],{},[1135,5058,5059],{"encoding":1137},"1≤j≤m",[1112,5061,5063,5082,5100],{"className":5062,"ariaHidden":1142},[1141],[1112,5064,5066,5070,5073,5076,5079],{"className":5065},[1146],[1112,5067],{"className":5068,"style":5069},[1150],"height:0.7804em;vertical-align:-0.136em;",[1112,5071,2765],{"className":5072},[1155],[1112,5074],{"className":5075,"style":1271},[1270],[1112,5077,5050],{"className":5078},[1275],[1112,5080],{"className":5081,"style":1271},[1270],[1112,5083,5085,5088,5091,5094,5097],{"className":5084},[1146],[1112,5086],{"className":5087,"style":4806},[1150],[1112,5089,4921],{"className":5090,"style":4972},[1155,1156],[1112,5092],{"className":5093,"style":1271},[1270],[1112,5095,5050],{"className":5096},[1275],[1112,5098],{"className":5099,"style":1271},[1270],[1112,5101,5103,5106],{"className":5102},[1146],[1112,5104],{"className":5105,"style":3092},[1150],[1112,5107,4764],{"className":5108},[1155,1156]," ​。",[2044,5111,5112,5113,5252,5253,5325],{},"如果 G 是有向图，则路径也是有向的，顶点序列应满足  ",[1112,5114,5116,5158],{"className":5115},[1115],[1112,5117,5119],{"className":5118},[1119],[1121,5120,5121],{"xmlns":1123},[1125,5122,5123,5155],{},[1128,5124,5125,5127,5129,5131,5133,5135,5137,5139,5141,5143,5145,5147,5149,5151,5153],{},[1235,5126,1778],{},[1131,5128,3001],{},[1131,5130,4729],{},[1235,5132,1246],{"separator":1142},[1131,5134,4921],{},[1235,5136,2761],{},[2763,5138,2765],{},[1235,5140,1246],{"separator":1142},[1131,5142,3001],{},[1131,5144,4729],{},[1235,5146,1246],{"separator":1142},[1131,5148,4921],{},[1235,5150,1789],{},[1235,5152,3012],{},[1131,5154,1203],{},[1135,5156,5157],{"encoding":1137},"\u003Cvi,j−1,vi,j>∈E",[1112,5159,5161,5173,5203,5243],{"className":5160,"ariaHidden":1142},[1141],[1112,5162,5164,5167,5170],{"className":5163},[1146],[1112,5165],{"className":5166,"style":1802},[1150],[1112,5168,1778],{"className":5169},[1275],[1112,5171],{"className":5172,"style":1271},[1270],[1112,5174,5176,5179,5182,5185,5188,5191,5194,5197,5200],{"className":5175},[1146],[1112,5177],{"className":5178,"style":4806},[1150],[1112,5180,3001],{"className":5181,"style":1828},[1155,1156],[1112,5183,4729],{"className":5184},[1155,1156],[1112,5186,1246],{"className":5187},[1296],[1112,5189],{"className":5190,"style":1300},[1270],[1112,5192,4921],{"className":5193,"style":4972},[1155,1156],[1112,5195],{"className":5196,"style":1187},[1270],[1112,5198,2761],{"className":5199},[2801],[1112,5201],{"className":5202,"style":1187},[1270],[1112,5204,5206,5209,5212,5215,5218,5221,5224,5227,5230,5233,5236,5240],{"className":5205},[1146],[1112,5207],{"className":5208,"style":4806},[1150],[1112,5210,2765],{"className":5211},[1155],[1112,5213,1246],{"className":5214},[1296],[1112,5216],{"className":5217,"style":1300},[1270],[1112,5219,3001],{"className":5220,"style":1828},[1155,1156],[1112,5222,4729],{"className":5223},[1155,1156],[1112,5225,1246],{"className":5226},[1296],[1112,5228],{"className":5229,"style":1300},[1270],[1112,5231,4921],{"className":5232,"style":4972},[1155,1156],[1112,5234],{"className":5235,"style":1271},[1270],[1112,5237,5239],{"className":5238},[1275],">∈",[1112,5241],{"className":5242,"style":1271},[1270],[1112,5244,5246,5249],{"className":5245},[1146],[1112,5247],{"className":5248,"style":1151},[1150],[1112,5250,1203],{"className":5251,"style":1218},[1155,1156]," ​， ",[1112,5254,5256,5277],{"className":5255},[1115],[1112,5257,5259],{"className":5258},[1119],[1121,5260,5261],{"xmlns":1123},[1125,5262,5263,5275],{},[1128,5264,5265,5267,5269,5271,5273],{},[2763,5266,2765],{},[1235,5268,5050],{},[1131,5270,4921],{},[1235,5272,5050],{},[1131,5274,4764],{},[1135,5276,5059],{"encoding":1137},[1112,5278,5280,5298,5316],{"className":5279,"ariaHidden":1142},[1141],[1112,5281,5283,5286,5289,5292,5295],{"className":5282},[1146],[1112,5284],{"className":5285,"style":5069},[1150],[1112,5287,2765],{"className":5288},[1155],[1112,5290],{"className":5291,"style":1271},[1270],[1112,5293,5050],{"className":5294},[1275],[1112,5296],{"className":5297,"style":1271},[1270],[1112,5299,5301,5304,5307,5310,5313],{"className":5300},[1146],[1112,5302],{"className":5303,"style":4806},[1150],[1112,5305,4921],{"className":5306,"style":4972},[1155,1156],[1112,5308],{"className":5309,"style":1271},[1270],[1112,5311,5050],{"className":5312},[1275],[1112,5314],{"className":5315,"style":1271},[1270],[1112,5317,5319,5322],{"className":5318},[1146],[1112,5320],{"className":5321,"style":3092},[1150],[1112,5323,4764],{"className":5324},[1155,1156]," ​。路径长度是一条路径上经过的边或弧的数目。",[2044,5327,5328],{},"回路或环：第一个顶点和最后一个顶点相同的路径称为回路或环。",[2044,5330,5331],{},"简单路径、简单回路或简单环：序列中顶点不重复出现的路径称为简单路径。除了第一个顶点和最后一个顶点之外，其余顶点不重复出现的回路，称为简单回路或简单环。",[2044,5333,5334,5335,5363,5364,4644,5392,5453,5454,2329,5482,5543,5544,5693,5694,5764,5765,5836,5837,5865],{},"连通、连通图和连通分量：在无向图 ",[1112,5336,5338,5351],{"className":5337},[1115],[1112,5339,5341],{"className":5340},[1119],[1121,5342,5343],{"xmlns":1123},[1125,5344,5345,5349],{},[1128,5346,5347],{},[1131,5348,1133],{},[1135,5350,1133],{"encoding":1137},[1112,5352,5354],{"className":5353,"ariaHidden":1142},[1141],[1112,5355,5357,5360],{"className":5356},[1146],[1112,5358],{"className":5359,"style":1151},[1150],[1112,5361,1133],{"className":5362},[1155,1156]," 中，如果从顶点 ",[1112,5365,5367,5380],{"className":5366},[1115],[1112,5368,5370],{"className":5369},[1119],[1121,5371,5372],{"xmlns":1123},[1125,5373,5374,5378],{},[1128,5375,5376],{},[1131,5377,3001],{},[1135,5379,3001],{"encoding":1137},[1112,5381,5383],{"className":5382,"ariaHidden":1142},[1141],[1112,5384,5386,5389],{"className":5385},[1146],[1112,5387],{"className":5388,"style":3092},[1150],[1112,5390,3001],{"className":5391,"style":1828},[1155,1156],[1112,5393,5395,5412],{"className":5394},[1115],[1112,5396,5398],{"className":5397},[1119],[1121,5399,5400],{"xmlns":1123},[1125,5401,5402,5410],{},[1128,5403,5404],{},[2343,5405,5406,5408],{},[1131,5407,3001],{},[1235,5409,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1135,5411,3116],{"encoding":1137},[1112,5413,5415],{"className":5414,"ariaHidden":1142},[1141],[1112,5416,5418,5421],{"className":5417},[1146],[1112,5419],{"className":5420,"style":2384},[1150],[1112,5422,5424,5427],{"className":5423},[1155],[1112,5425,3001],{"className":5426,"style":1828},[1155,1156],[1112,5428,5430],{"className":5429},[2394],[1112,5431,5433],{"className":5432},[2398],[1112,5434,5436],{"className":5435},[2402],[1112,5437,5439],{"className":5438,"style":2384},[2406],[1112,5440,5441,5444],{"style":2409},[1112,5442],{"className":5443,"style":2414},[2413],[1112,5445,5447],{"className":5446},[2418,2419,2420,2421],[1112,5448,5450],{"className":5449},[1155,2421],[1112,5451,2351],{"className":5452},[1155,2421]," 有路径，则称 ",[1112,5455,5457,5470],{"className":5456},[1115],[1112,5458,5460],{"className":5459},[1119],[1121,5461,5462],{"xmlns":1123},[1125,5463,5464,5468],{},[1128,5465,5466],{},[1131,5467,3001],{},[1135,5469,3001],{"encoding":1137},[1112,5471,5473],{"className":5472,"ariaHidden":1142},[1141],[1112,5474,5476,5479],{"className":5475},[1146],[1112,5477],{"className":5478,"style":3092},[1150],[1112,5480,3001],{"className":5481,"style":1828},[1155,1156],[1112,5483,5485,5502],{"className":5484},[1115],[1112,5486,5488],{"className":5487},[1119],[1121,5489,5490],{"xmlns":1123},[1125,5491,5492,5500],{},[1128,5493,5494],{},[2343,5495,5496,5498],{},[1131,5497,3001],{},[1235,5499,2351],{"mathvariant":2349,"lspace":2350,"rspace":2350},[1135,5501,3116],{"encoding":1137},[1112,5503,5505],{"className":5504,"ariaHidden":1142},[1141],[1112,5506,5508,5511],{"className":5507},[1146],[1112,5509],{"className":5510,"style":2384},[1150],[1112,5512,5514,5517],{"className":5513},[1155],[1112,5515,3001],{"className":5516,"style":1828},[1155,1156],[1112,5518,5520],{"className":5519},[2394],[1112,5521,5523],{"className":5522},[2398],[1112,5524,5526],{"className":5525},[2402],[1112,5527,5529],{"className":5528,"style":2384},[2406],[1112,5530,5531,5534],{"style":2409},[1112,5532],{"className":5533,"style":2414},[2413],[1112,5535,5537],{"className":5536},[2418,2419,2420,2421],[1112,5538,5540],{"className":5539},[1155,2421],[1112,5541,2351],{"className":5542},[1155,2421]," 是连通的。如果对于图中任意两个顶点 ​ ",[1112,5545,5547,5577],{"className":5546},[1115],[1112,5548,5550],{"className":5549},[1119],[1121,5551,5552],{"xmlns":1123},[1125,5553,5554,5574],{},[1128,5555,5556,5562,5564,5570,5572],{},[3638,5557,5558,5560],{},[1131,5559,3001],{},[1131,5561,4729],{},[1235,5563,1246],{"separator":1142},[3638,5565,5566,5568],{},[1131,5567,3001],{},[1131,5569,4921],{},[1235,5571,3012],{},[1131,5573,1172],{},[1135,5575,5576],{"encoding":1137},"v_i,v_j∈V",[1112,5578,5580,5684],{"className":5579,"ariaHidden":1142},[1141],[1112,5581,5583,5587,5628,5631,5634,5675,5678,5681],{"className":5582},[1146],[1112,5584],{"className":5585,"style":5586},[1150],"height:0.8252em;vertical-align:-0.2861em;",[1112,5588,5590,5593],{"className":5589},[1155],[1112,5591,3001],{"className":5592,"style":1828},[1155,1156],[1112,5594,5596],{"className":5595},[2394],[1112,5597,5599,5620],{"className":5598},[2398,3669],[1112,5600,5602,5617],{"className":5601},[2402],[1112,5603,5606],{"className":5604,"style":5605},[2406],"height:0.3117em;",[1112,5607,5608,5611],{"style":3679},[1112,5609],{"className":5610,"style":2414},[2413],[1112,5612,5614],{"className":5613},[2418,2419,2420,2421],[1112,5615,4729],{"className":5616},[1155,1156,2421],[1112,5618,3693],{"className":5619},[3692],[1112,5621,5623],{"className":5622},[2402],[1112,5624,5626],{"className":5625,"style":3700},[2406],[1112,5627],{},[1112,5629,1246],{"className":5630},[1296],[1112,5632],{"className":5633,"style":1300},[1270],[1112,5635,5637,5640],{"className":5636},[1155],[1112,5638,3001],{"className":5639,"style":1828},[1155,1156],[1112,5641,5643],{"className":5642},[2394],[1112,5644,5646,5666],{"className":5645},[2398,3669],[1112,5647,5649,5663],{"className":5648},[2402],[1112,5650,5652],{"className":5651,"style":5605},[2406],[1112,5653,5654,5657],{"style":3679},[1112,5655],{"className":5656,"style":2414},[2413],[1112,5658,5660],{"className":5659},[2418,2419,2420,2421],[1112,5661,4921],{"className":5662,"style":4972},[1155,1156,2421],[1112,5664,3693],{"className":5665},[3692],[1112,5667,5669],{"className":5668},[2402],[1112,5670,5673],{"className":5671,"style":5672},[2406],"height:0.2861em;",[1112,5674],{},[1112,5676],{"className":5677,"style":1271},[1270],[1112,5679,3012],{"className":5680},[1275],[1112,5682],{"className":5683,"style":1271},[1270],[1112,5685,5687,5690],{"className":5686},[1146],[1112,5688],{"className":5689,"style":1151},[1150],[1112,5691,1172],{"className":5692,"style":1187},[1155,1156]," ，​ ",[1112,5695,5697,5715],{"className":5696},[1115],[1112,5698,5700],{"className":5699},[1119],[1121,5701,5702],{"xmlns":1123},[1125,5703,5704,5712],{},[1128,5705,5706],{},[3638,5707,5708,5710],{},[1131,5709,3001],{},[1131,5711,4729],{},[1135,5713,5714],{"encoding":1137},"v_i",[1112,5716,5718],{"className":5717,"ariaHidden":1142},[1141],[1112,5719,5721,5724],{"className":5720},[1146],[1112,5722],{"className":5723,"style":3656},[1150],[1112,5725,5727,5730],{"className":5726},[1155],[1112,5728,3001],{"className":5729,"style":1828},[1155,1156],[1112,5731,5733],{"className":5732},[2394],[1112,5734,5736,5756],{"className":5735},[2398,3669],[1112,5737,5739,5753],{"className":5738},[2402],[1112,5740,5742],{"className":5741,"style":5605},[2406],[1112,5743,5744,5747],{"style":3679},[1112,5745],{"className":5746,"style":2414},[2413],[1112,5748,5750],{"className":5749},[2418,2419,2420,2421],[1112,5751,4729],{"className":5752},[1155,1156,2421],[1112,5754,3693],{"className":5755},[3692],[1112,5757,5759],{"className":5758},[2402],[1112,5760,5762],{"className":5761,"style":3700},[2406],[1112,5763],{},"  和  ",[1112,5766,5768,5786],{"className":5767},[1115],[1112,5769,5771],{"className":5770},[1119],[1121,5772,5773],{"xmlns":1123},[1125,5774,5775,5783],{},[1128,5776,5777],{},[3638,5778,5779,5781],{},[1131,5780,3001],{},[1131,5782,4921],{},[1135,5784,5785],{"encoding":1137},"v_j",[1112,5787,5789],{"className":5788,"ariaHidden":1142},[1141],[1112,5790,5792,5796],{"className":5791},[1146],[1112,5793],{"className":5794,"style":5795},[1150],"height:0.7167em;vertical-align:-0.2861em;",[1112,5797,5799,5802],{"className":5798},[1155],[1112,5800,3001],{"className":5801,"style":1828},[1155,1156],[1112,5803,5805],{"className":5804},[2394],[1112,5806,5808,5828],{"className":5807},[2398,3669],[1112,5809,5811,5825],{"className":5810},[2402],[1112,5812,5814],{"className":5813,"style":5605},[2406],[1112,5815,5816,5819],{"style":3679},[1112,5817],{"className":5818,"style":2414},[2413],[1112,5820,5822],{"className":5821},[2418,2419,2420,2421],[1112,5823,4921],{"className":5824,"style":4972},[1155,1156,2421],[1112,5826,3693],{"className":5827},[3692],[1112,5829,5831],{"className":5830},[2402],[1112,5832,5834],{"className":5833,"style":5672},[2406],[1112,5835],{}," ​ 都是连通的，则称 ",[1112,5838,5840,5853],{"className":5839},[1115],[1112,5841,5843],{"className":5842},[1119],[1121,5844,5845],{"xmlns":1123},[1125,5846,5847,5851],{},[1128,5848,5849],{},[1131,5850,1133],{},[1135,5852,1133],{"encoding":1137},[1112,5854,5856],{"className":5855,"ariaHidden":1142},[1141],[1112,5857,5859,5862],{"className":5858},[1146],[1112,5860],{"className":5861,"style":1151},[1150],[1112,5863,1133],{"className":5864},[1155,1156]," 是连通图。而所谓连通分量，指的是无向图中的极大连通子图。例如，下图（a）就是一个连通图，而图（b）则是非连通图，但它有 3 个连通分量，见图（c）。",[1091,5867,5868],{},[1634,5869],{"alt":2039,"src":5870},"https:\u002F\u002Fpica.zhimg.com\u002F80\u002Fv2-222e4b072a43944cb3adaedcdf85c920_720w.webp",[2041,5872,5873],{},[2044,5874,5875,5876,5904,5905,5693,6050,6224,6225,6294,6295,6364,6365,6294,6434,6503,6504,6532],{},"强连通图和强连通分量：在有向图 ",[1112,5877,5879,5892],{"className":5878},[1115],[1112,5880,5882],{"className":5881},[1119],[1121,5883,5884],{"xmlns":1123},[1125,5885,5886,5890],{},[1128,5887,5888],{},[1131,5889,1133],{},[1135,5891,1133],{"encoding":1137},[1112,5893,5895],{"className":5894,"ariaHidden":1142},[1141],[1112,5896,5898,5901],{"className":5897},[1146],[1112,5899],{"className":5900,"style":1151},[1150],[1112,5902,1133],{"className":5903},[1155,1156]," 中，如果对于每一对 ​ ",[1112,5906,5908,5937],{"className":5907},[1115],[1112,5909,5911],{"className":5910},[1119],[1121,5912,5913],{"xmlns":1123},[1125,5914,5915,5935],{},[1128,5916,5917,5923,5925,5931,5933],{},[3638,5918,5919,5921],{},[1131,5920,3001],{},[1131,5922,4729],{},[1235,5924,1246],{"separator":1142},[3638,5926,5927,5929],{},[1131,5928,3001],{},[1131,5930,4921],{},[1235,5932,3012],{},[1131,5934,1172],{},[1135,5936,5576],{"encoding":1137},[1112,5938,5940,6041],{"className":5939,"ariaHidden":1142},[1141],[1112,5941,5943,5946,5986,5989,5992,6032,6035,6038],{"className":5942},[1146],[1112,5944],{"className":5945,"style":5586},[1150],[1112,5947,5949,5952],{"className":5948},[1155],[1112,5950,3001],{"className":5951,"style":1828},[1155,1156],[1112,5953,5955],{"className":5954},[2394],[1112,5956,5958,5978],{"className":5957},[2398,3669],[1112,5959,5961,5975],{"className":5960},[2402],[1112,5962,5964],{"className":5963,"style":5605},[2406],[1112,5965,5966,5969],{"style":3679},[1112,5967],{"className":5968,"style":2414},[2413],[1112,5970,5972],{"className":5971},[2418,2419,2420,2421],[1112,5973,4729],{"className":5974},[1155,1156,2421],[1112,5976,3693],{"className":5977},[3692],[1112,5979,5981],{"className":5980},[2402],[1112,5982,5984],{"className":5983,"style":3700},[2406],[1112,5985],{},[1112,5987,1246],{"className":5988},[1296],[1112,5990],{"className":5991,"style":1300},[1270],[1112,5993,5995,5998],{"className":5994},[1155],[1112,5996,3001],{"className":5997,"style":1828},[1155,1156],[1112,5999,6001],{"className":6000},[2394],[1112,6002,6004,6024],{"className":6003},[2398,3669],[1112,6005,6007,6021],{"className":6006},[2402],[1112,6008,6010],{"className":6009,"style":5605},[2406],[1112,6011,6012,6015],{"style":3679},[1112,6013],{"className":6014,"style":2414},[2413],[1112,6016,6018],{"className":6017},[2418,2419,2420,2421],[1112,6019,4921],{"className":6020,"style":4972},[1155,1156,2421],[1112,6022,3693],{"className":6023},[3692],[1112,6025,6027],{"className":6026},[2402],[1112,6028,6030],{"className":6029,"style":5672},[2406],[1112,6031],{},[1112,6033],{"className":6034,"style":1271},[1270],[1112,6036,3012],{"className":6037},[1275],[1112,6039],{"className":6040,"style":1271},[1270],[1112,6042,6044,6047],{"className":6043},[1146],[1112,6045],{"className":6046,"style":1151},[1150],[1112,6048,1172],{"className":6049,"style":1187},[1155,1156],[1112,6051,6053,6080],{"className":6052},[1115],[1112,6054,6056],{"className":6055},[1119],[1121,6057,6058],{"xmlns":1123},[1125,6059,6060,6077],{},[1128,6061,6062,6068,6071],{},[3638,6063,6064,6066],{},[1131,6065,3001],{},[1131,6067,4729],{},[1235,6069,6070],{"mathvariant":2349},"≠",[3638,6072,6073,6075],{},[1131,6074,3001],{},[1131,6076,4921],{},[1135,6078,6079],{"encoding":1137},"v_i≠v_j",[1112,6081,6083,6178],{"className":6082,"ariaHidden":1142},[1141],[1112,6084,6086,6089,6129,6132,6175],{"className":6085},[1146],[1112,6087],{"className":6088,"style":2957},[1150],[1112,6090,6092,6095],{"className":6091},[1155],[1112,6093,3001],{"className":6094,"style":1828},[1155,1156],[1112,6096,6098],{"className":6097},[2394],[1112,6099,6101,6121],{"className":6100},[2398,3669],[1112,6102,6104,6118],{"className":6103},[2402],[1112,6105,6107],{"className":6106,"style":5605},[2406],[1112,6108,6109,6112],{"style":3679},[1112,6110],{"className":6111,"style":2414},[2413],[1112,6113,6115],{"className":6114},[2418,2419,2420,2421],[1112,6116,4729],{"className":6117},[1155,1156,2421],[1112,6119,3693],{"className":6120},[3692],[1112,6122,6124],{"className":6123},[2402],[1112,6125,6127],{"className":6126,"style":3700},[2406],[1112,6128],{},[1112,6130],{"className":6131,"style":1271},[1270],[1112,6133,6135,6168,6172],{"className":6134},[1275],[1112,6136,6138],{"className":6137},[1275],[1112,6139,6142],{"className":6140},[1155,6141],"vbox",[1112,6143,6146],{"className":6144},[6145],"thinbox",[1112,6147,6150,6153,6164],{"className":6148},[6149],"rlap",[1112,6151],{"className":6152,"style":2957},[1150],[1112,6154,6157],{"className":6155},[6156],"inner",[1112,6158,6160],{"className":6159},[1155],[1112,6161,6163],{"className":6162},[1275],"",[1112,6165],{"className":6166},[6167],"fix",[1112,6169],{"className":6170},[1270,6171],"nobreak",[1112,6173,1237],{"className":6174},[1275],[1112,6176],{"className":6177,"style":1271},[1270],[1112,6179,6181,6184],{"className":6180},[1146],[1112,6182],{"className":6183,"style":5795},[1150],[1112,6185,6187,6190],{"className":6186},[1155],[1112,6188,3001],{"className":6189,"style":1828},[1155,1156],[1112,6191,6193],{"className":6192},[2394],[1112,6194,6196,6216],{"className":6195},[2398,3669],[1112,6197,6199,6213],{"className":6198},[2402],[1112,6200,6202],{"className":6201,"style":5605},[2406],[1112,6203,6204,6207],{"style":3679},[1112,6205],{"className":6206,"style":2414},[2413],[1112,6208,6210],{"className":6209},[2418,2419,2420,2421],[1112,6211,4921],{"className":6212,"style":4972},[1155,1156,2421],[1112,6214,3693],{"className":6215},[3692],[1112,6217,6219],{"className":6218},[2402],[1112,6220,6222],{"className":6221,"style":5672},[2406],[1112,6223],{}," ，从 ​ ",[1112,6226,6228,6245],{"className":6227},[1115],[1112,6229,6231],{"className":6230},[1119],[1121,6232,6233],{"xmlns":1123},[1125,6234,6235,6243],{},[1128,6236,6237],{},[3638,6238,6239,6241],{},[1131,6240,3001],{},[1131,6242,4729],{},[1135,6244,5714],{"encoding":1137},[1112,6246,6248],{"className":6247,"ariaHidden":1142},[1141],[1112,6249,6251,6254],{"className":6250},[1146],[1112,6252],{"className":6253,"style":3656},[1150],[1112,6255,6257,6260],{"className":6256},[1155],[1112,6258,3001],{"className":6259,"style":1828},[1155,1156],[1112,6261,6263],{"className":6262},[2394],[1112,6264,6266,6286],{"className":6265},[2398,3669],[1112,6267,6269,6283],{"className":6268},[2402],[1112,6270,6272],{"className":6271,"style":5605},[2406],[1112,6273,6274,6277],{"style":3679},[1112,6275],{"className":6276,"style":2414},[2413],[1112,6278,6280],{"className":6279},[2418,2419,2420,2421],[1112,6281,4729],{"className":6282},[1155,1156,2421],[1112,6284,3693],{"className":6285},[3692],[1112,6287,6289],{"className":6288},[2402],[1112,6290,6292],{"className":6291,"style":3700},[2406],[1112,6293],{},"  到 ​ ",[1112,6296,6298,6315],{"className":6297},[1115],[1112,6299,6301],{"className":6300},[1119],[1121,6302,6303],{"xmlns":1123},[1125,6304,6305,6313],{},[1128,6306,6307],{},[3638,6308,6309,6311],{},[1131,6310,3001],{},[1131,6312,4921],{},[1135,6314,5785],{"encoding":1137},[1112,6316,6318],{"className":6317,"ariaHidden":1142},[1141],[1112,6319,6321,6324],{"className":6320},[1146],[1112,6322],{"className":6323,"style":5795},[1150],[1112,6325,6327,6330],{"className":6326},[1155],[1112,6328,3001],{"className":6329,"style":1828},[1155,1156],[1112,6331,6333],{"className":6332},[2394],[1112,6334,6336,6356],{"className":6335},[2398,3669],[1112,6337,6339,6353],{"className":6338},[2402],[1112,6340,6342],{"className":6341,"style":5605},[2406],[1112,6343,6344,6347],{"style":3679},[1112,6345],{"className":6346,"style":2414},[2413],[1112,6348,6350],{"className":6349},[2418,2419,2420,2421],[1112,6351,4921],{"className":6352,"style":4972},[1155,1156,2421],[1112,6354,3693],{"className":6355},[3692],[1112,6357,6359],{"className":6358},[2402],[1112,6360,6362],{"className":6361,"style":5672},[2406],[1112,6363],{},"  和从 ​ ",[1112,6366,6368,6385],{"className":6367},[1115],[1112,6369,6371],{"className":6370},[1119],[1121,6372,6373],{"xmlns":1123},[1125,6374,6375,6383],{},[1128,6376,6377],{},[3638,6378,6379,6381],{},[1131,6380,3001],{},[1131,6382,4921],{},[1135,6384,5785],{"encoding":1137},[1112,6386,6388],{"className":6387,"ariaHidden":1142},[1141],[1112,6389,6391,6394],{"className":6390},[1146],[1112,6392],{"className":6393,"style":5795},[1150],[1112,6395,6397,6400],{"className":6396},[1155],[1112,6398,3001],{"className":6399,"style":1828},[1155,1156],[1112,6401,6403],{"className":6402},[2394],[1112,6404,6406,6426],{"className":6405},[2398,3669],[1112,6407,6409,6423],{"className":6408},[2402],[1112,6410,6412],{"className":6411,"style":5605},[2406],[1112,6413,6414,6417],{"style":3679},[1112,6415],{"className":6416,"style":2414},[2413],[1112,6418,6420],{"className":6419},[2418,2419,2420,2421],[1112,6421,4921],{"className":6422,"style":4972},[1155,1156,2421],[1112,6424,3693],{"className":6425},[3692],[1112,6427,6429],{"className":6428},[2402],[1112,6430,6432],{"className":6431,"style":5672},[2406],[1112,6433],{},[1112,6435,6437,6454],{"className":6436},[1115],[1112,6438,6440],{"className":6439},[1119],[1121,6441,6442],{"xmlns":1123},[1125,6443,6444,6452],{},[1128,6445,6446],{},[3638,6447,6448,6450],{},[1131,6449,3001],{},[1131,6451,4729],{},[1135,6453,5714],{"encoding":1137},[1112,6455,6457],{"className":6456,"ariaHidden":1142},[1141],[1112,6458,6460,6463],{"className":6459},[1146],[1112,6461],{"className":6462,"style":3656},[1150],[1112,6464,6466,6469],{"className":6465},[1155],[1112,6467,3001],{"className":6468,"style":1828},[1155,1156],[1112,6470,6472],{"className":6471},[2394],[1112,6473,6475,6495],{"className":6474},[2398,3669],[1112,6476,6478,6492],{"className":6477},[2402],[1112,6479,6481],{"className":6480,"style":5605},[2406],[1112,6482,6483,6486],{"style":3679},[1112,6484],{"className":6485,"style":2414},[2413],[1112,6487,6489],{"className":6488},[2418,2419,2420,2421],[1112,6490,4729],{"className":6491},[1155,1156,2421],[1112,6493,3693],{"className":6494},[3692],[1112,6496,6498],{"className":6497},[2402],[1112,6499,6501],{"className":6500,"style":3700},[2406],[1112,6502],{},"  都存在路径，则称 ",[1112,6505,6507,6520],{"className":6506},[1115],[1112,6508,6510],{"className":6509},[1119],[1121,6511,6512],{"xmlns":1123},[1125,6513,6514,6518],{},[1128,6515,6516],{},[1131,6517,1133],{},[1135,6519,1133],{"encoding":1137},[1112,6521,6523],{"className":6522,"ariaHidden":1142},[1141],[1112,6524,6526,6529],{"className":6525},[1146],[1112,6527],{"className":6528,"style":1151},[1150],[1112,6530,1133],{"className":6531},[1155,1156]," 是强连通图。有向图中的极大强连通子图称作有向图的强连通分量。例如，下图（a）就是一个强连通图，而图（b）则不是强连通图，但它有两个强连通分量，见图（c）。",[1091,6534,6535],{},[1634,6536],{"alt":2039,"src":6537},"https:\u002F\u002Fpic1.zhimg.com\u002F80\u002Fv2-6e6d3ca5a9c049e494b224765ceb1b1c_720w.webp",[2041,6539,6540],{},[2044,6541,6542],{},"连通图的生成树：一个极小连通子图，它含有图中全部顶点，但只有足以构成一棵树的 n-1 边，这样的连通子图称为连通图的生成树。例如，下图（c）是图（a）的最大连通分量的一棵生成树。如果在一棵生成树上添加一条边，必定构成一个环，因为这条边使得它依附的那两个顶点之间有了第二条路径。",[1091,6544,6545],{},[1634,6546],{"alt":2039,"src":6547},"https:\u002F\u002Fpic4.zhimg.com\u002F80\u002Fv2-4592e9e0746070d2501fddda30a5a65d_720w.webp",[2041,6549,6550],{},[2044,6551,6552],{},"有向树和生成森林：有一个顶点的入度为 0，其余顶点的入度均为 1 的有向图称为有向树。一个有向图的生成森林由若干棵有向树组成，含有图中全部顶点，但只有足以构成若干棵不相交的有向树的弧。例如，下图（a）是一棵有向树，图（b）是一个有向图，它的森林是图（c）。",[1091,6554,6555],{},[1634,6556],{"alt":2039,"src":6557},"https:\u002F\u002Fpic1.zhimg.com\u002F80\u002Fv2-2f72ebb4263a60bc600e7a48aa9f501c_720w.webp",[1101,6559,6561],{"id":6560},"三存储结构",[1105,6562,6563],{},"三、存储结构",[1091,6565,6566,6567,6574],{},"由于图的结构比较复杂，任意两个顶点之间都可能存在联系，因此无法以数据元素在存储区中的物理位置来表示元素之间的关系，即图没有顺序存储结构，但可以借助二维数组来表示元素之间的关系，即",[6568,6569,6573],"a",{"href":6570,"rel":6571},"https:\u002F\u002Fzhida.zhihu.com\u002Fsearch?content_id=240065349&content_type=Article&match_order=1&q=%E9%82%BB%E6%8E%A5%E7%9F%A9%E9%98%B5&zhida_source=entity",[6572],"nofollow","邻接矩阵","表示法。",[1091,6576,6577,6578,6583,6584,1188,6589,6594],{},"另一方面，由于图的任意两个顶点间都可能存在关系，因此，用链式存储表示图是很自然的事，图的链式存储有多种，有",[6568,6579,6582],{"href":6580,"rel":6581},"https:\u002F\u002Fzhida.zhihu.com\u002Fsearch?content_id=240065349&content_type=Article&match_order=1&q=%E9%82%BB%E6%8E%A5%E8%A1%A8&zhida_source=entity",[6572],"邻接表","、",[6568,6585,6588],{"href":6586,"rel":6587},"https:\u002F\u002Fzhida.zhihu.com\u002Fsearch?content_id=240065349&content_type=Article&match_order=1&q=%E5%8D%81%E5%AD%97%E9%93%BE%E8%A1%A8&zhida_source=entity",[6572],"十字链表",[6568,6590,6593],{"href":6591,"rel":6592},"https:\u002F\u002Fzhida.zhihu.com\u002Fsearch?content_id=240065349&content_type=Article&match_order=1&q=%E9%82%BB%E6%8E%A5%E5%A4%9A%E9%87%8D%E8%A1%A8&zhida_source=entity",[6572],"邻接多重表","，应根据实际需要的不同选择不同的存储结构。",[6596,6597,6599],"h3",{"id":6598},"_31邻接矩阵",[1105,6600,6601],{},"3.1、邻接矩阵",[6596,6603,6605],{"id":6604},"_311表示法",[1105,6606,6607],{},"3.1.1、表示法",[1091,6609,6610,6611,6668,6669,6697],{},"邻接矩阵（Adjacency Matrix）是表示顶点之间相邻关系的矩阵。设 ",[1112,6612,6614,6638],{"className":6613},[1115],[1112,6615,6617],{"className":6616},[1119],[1121,6618,6619],{"xmlns":1123},[1125,6620,6621,6635],{},[1128,6622,6623,6625,6627,6629,6631,6633],{},[1131,6624,1133],{},[1235,6626,1241],{"stretchy":1240},[1131,6628,1172],{},[1235,6630,1246],{"separator":1142},[1131,6632,1203],{},[1235,6634,1251],{"stretchy":1240},[1135,6636,6637],{"encoding":1137},"G(V, E)",[1112,6639,6641],{"className":6640,"ariaHidden":1142},[1141],[1112,6642,6644,6647,6650,6653,6656,6659,6662,6665],{"className":6643},[1146],[1112,6645],{"className":6646,"style":1285},[1150],[1112,6648,1133],{"className":6649},[1155,1156],[1112,6651,1241],{"className":6652},[1289],[1112,6654,1172],{"className":6655,"style":1187},[1155,1156],[1112,6657,1246],{"className":6658},[1296],[1112,6660],{"className":6661,"style":1300},[1270],[1112,6663,1203],{"className":6664,"style":1218},[1155,1156],[1112,6666,1251],{"className":6667},[1307],"是具有 n 个顶点的图，则 ",[1112,6670,6672,6685],{"className":6671},[1115],[1112,6673,6675],{"className":6674},[1119],[1121,6676,6677],{"xmlns":1123},[1125,6678,6679,6683],{},[1128,6680,6681],{},[1131,6682,1133],{},[1135,6684,1133],{"encoding":1137},[1112,6686,6688],{"className":6687,"ariaHidden":1142},[1141],[1112,6689,6691,6694],{"className":6690},[1146],[1112,6692],{"className":6693,"style":1151},[1150],[1112,6695,1133],{"className":6696},[1155,1156]," 的邻接矩阵是具有如下性质的 n 阶方阵：",[1091,6699,6700],{},[1112,6701,6703,6817],{"className":6702},[1115],[1112,6704,6706],{"className":6705},[1119],[1121,6707,6708],{"xmlns":1123},[1125,6709,6710,6814],{},[1128,6711,6712,6715,6718,6720,6723,6725,6727,6729,6731],{},[1131,6713,6714],{},"A",[1235,6716,6717],{"stretchy":1240},"[",[1131,6719,4729],{},[1235,6721,6722],{"stretchy":1240},"]",[1235,6724,6717],{"stretchy":1240},[1131,6726,4921],{},[1235,6728,6722],{"stretchy":1240},[1235,6730,1237],{},[1128,6732,6733,6736],{},[1235,6734,6735],{"fence":1142},"{",[6737,6738,6742,6800],"mtable",{"rowspacing":6739,"columnalign":6740,"columnspacing":6741},"0.36em","left left","1em",[6743,6744,6745,6753],"mtr",{},[6746,6747,6748],"mtd",{},[6749,6750,6751],"mstyle",{"scriptlevel":4734,"displaystyle":1240},[2763,6752,2765],{},[6746,6754,6755],{},[6749,6756,6757],{"scriptlevel":4734,"displaystyle":1240},[1128,6758,6759,6761,6767,6769,6775,6778,6780,6786,6788,6794,6796,6798],{},[2536,6760,1778],{},[3638,6762,6763,6765],{},[1131,6764,3001],{},[1131,6766,4729],{},[1235,6768,1246],{"separator":1142},[3638,6770,6771,6773],{},[1131,6772,3001],{},[1131,6774,4921],{},[2536,6776,6777],{},"> 或 ",[1235,6779,1241],{"stretchy":1240},[3638,6781,6782,6784],{},[1131,6783,3001],{},[1131,6785,4729],{},[1235,6787,1246],{"separator":1142},[3638,6789,6790,6792],{},[1131,6791,3001],{},[1131,6793,4921],{},[1235,6795,1251],{"stretchy":1240},[1235,6797,3012],{},[1131,6799,1203],{},[6743,6801,6802,6808],{},[6746,6803,6804],{},[6749,6805,6806],{"scriptlevel":4734,"displaystyle":1240},[2763,6807,4734],{},[6746,6809,6810],{},[6749,6811,6812],{"scriptlevel":4734,"displaystyle":1240},[2536,6813,12],{},[1135,6815,6816],{"encoding":1137},"A[i][j] = \\begin{cases} 1 & \\text{\u003C}v_i, v_j\\text{>} \\text{ 或 } (v_i, v_j) \\in E \\\\ 0 & \\text{其他} \\end{cases}",[1112,6818,6820,6856],{"className":6819,"ariaHidden":1142},[1141],[1112,6821,6823,6826,6829,6832,6835,6838,6841,6844,6847,6850,6853],{"className":6822},[1146],[1112,6824],{"className":6825,"style":1285},[1150],[1112,6827,6714],{"className":6828},[1155,1156],[1112,6830,6717],{"className":6831},[1289],[1112,6833,4729],{"className":6834},[1155,1156],[1112,6836,6722],{"className":6837},[1307],[1112,6839,6717],{"className":6840},[1289],[1112,6842,4921],{"className":6843,"style":4972},[1155,1156],[1112,6845,6722],{"className":6846},[1307],[1112,6848],{"className":6849,"style":1271},[1270],[1112,6851,1237],{"className":6852},[1275],[1112,6854],{"className":6855,"style":1271},[1270],[1112,6857,6859,6863],{"className":6858},[1146],[1112,6860],{"className":6861,"style":6862},[1150],"height:3em;vertical-align:-1.25em;",[1112,6864,6867,6877,7200],{"className":6865},[6866],"minner",[1112,6868,6872],{"className":6869,"style":6871},[1289,6870],"delimcenter","top:0em;",[1112,6873,6735],{"className":6874},[6875,6876],"delimsizing","size4",[1112,6878,6880],{"className":6879},[1155],[1112,6881,6883,6934,6939],{"className":6882},[6737],[1112,6884,6887],{"className":6885},[6886],"col-align-l",[1112,6888,6890,6925],{"className":6889},[2398,3669],[1112,6891,6893,6922],{"className":6892},[2402],[1112,6894,6897,6910],{"className":6895,"style":6896},[2406],"height:1.69em;",[1112,6898,6900,6904],{"style":6899},"top:-3.69em;",[1112,6901],{"className":6902,"style":6903},[2413],"height:3.008em;",[1112,6905,6907],{"className":6906},[1155],[1112,6908,2765],{"className":6909},[1155],[1112,6911,6913,6916],{"style":6912},"top:-2.25em;",[1112,6914],{"className":6915,"style":6903},[2413],[1112,6917,6919],{"className":6918},[1155],[1112,6920,4734],{"className":6921},[1155],[1112,6923,3693],{"className":6924},[3692],[1112,6926,6928],{"className":6927},[2402],[1112,6929,6932],{"className":6930,"style":6931},[2406],"height:1.19em;",[1112,6933],{},[1112,6935],{"className":6936,"style":6938},[6937],"arraycolsep","width:1em;",[1112,6940,6942],{"className":6941},[6886],[1112,6943,6945,7192],{"className":6944},[2398,3669],[1112,6946,6948,7189],{"className":6947},[2402],[1112,6949,6951,7175],{"className":6950,"style":6896},[2406],[1112,6952,6953,6956],{"style":6899},[1112,6954],{"className":6955,"style":6903},[2413],[1112,6957,6959,6965,7005,7008,7011,7051,7057,7071,7074,7114,7117,7120,7160,7163,7166,7169,7172],{"className":6958},[1155],[1112,6960,6962],{"className":6961},[1155,2035],[1112,6963,1778],{"className":6964},[1155],[1112,6966,6968,6971],{"className":6967},[1155],[1112,6969,3001],{"className":6970,"style":1828},[1155,1156],[1112,6972,6974],{"className":6973},[2394],[1112,6975,6977,6997],{"className":6976},[2398,3669],[1112,6978,6980,6994],{"className":6979},[2402],[1112,6981,6983],{"className":6982,"style":5605},[2406],[1112,6984,6985,6988],{"style":3679},[1112,6986],{"className":6987,"style":2414},[2413],[1112,6989,6991],{"className":6990},[2418,2419,2420,2421],[1112,6992,4729],{"className":6993},[1155,1156,2421],[1112,6995,3693],{"className":6996},[3692],[1112,6998,7000],{"className":6999},[2402],[1112,7001,7003],{"className":7002,"style":3700},[2406],[1112,7004],{},[1112,7006,1246],{"className":7007},[1296],[1112,7009],{"className":7010,"style":1300},[1270],[1112,7012,7014,7017],{"className":7013},[1155],[1112,7015,3001],{"className":7016,"style":1828},[1155,1156],[1112,7018,7020],{"className":7019},[2394],[1112,7021,7023,7043],{"className":7022},[2398,3669],[1112,7024,7026,7040],{"className":7025},[2402],[1112,7027,7029],{"className":7028,"style":5605},[2406],[1112,7030,7031,7034],{"style":3679},[1112,7032],{"className":7033,"style":2414},[2413],[1112,7035,7037],{"className":7036},[2418,2419,2420,2421],[1112,7038,4921],{"className":7039,"style":4972},[1155,1156,2421],[1112,7041,3693],{"className":7042},[3692],[1112,7044,7046],{"className":7045},[2402],[1112,7047,7049],{"className":7048,"style":5672},[2406],[1112,7050],{},[1112,7052,7054],{"className":7053},[1155,2035],[1112,7055,1789],{"className":7056},[1155],[1112,7058,7060,7063,7068],{"className":7059},[1155,2035],[1112,7061,4048],{"className":7062},[1155],[1112,7064,7067],{"className":7065},[1155,7066],"cjk_fallback","或",[1112,7069,4048],{"className":7070},[1155],[1112,7072,1241],{"className":7073},[1289],[1112,7075,7077,7080],{"className":7076},[1155],[1112,7078,3001],{"className":7079,"style":1828},[1155,1156],[1112,7081,7083],{"className":7082},[2394],[1112,7084,7086,7106],{"className":7085},[2398,3669],[1112,7087,7089,7103],{"className":7088},[2402],[1112,7090,7092],{"className":7091,"style":5605},[2406],[1112,7093,7094,7097],{"style":3679},[1112,7095],{"className":7096,"style":2414},[2413],[1112,7098,7100],{"className":7099},[2418,2419,2420,2421],[1112,7101,4729],{"className":7102},[1155,1156,2421],[1112,7104,3693],{"className":7105},[3692],[1112,7107,7109],{"className":7108},[2402],[1112,7110,7112],{"className":7111,"style":3700},[2406],[1112,7113],{},[1112,7115,1246],{"className":7116},[1296],[1112,7118],{"className":7119,"style":1300},[1270],[1112,7121,7123,7126],{"className":7122},[1155],[1112,7124,3001],{"className":7125,"style":1828},[1155,1156],[1112,7127,7129],{"className":7128},[2394],[1112,7130,7132,7152],{"className":7131},[2398,3669],[1112,7133,7135,7149],{"className":7134},[2402],[1112,7136,7138],{"className":7137,"style":5605},[2406],[1112,7139,7140,7143],{"style":3679},[1112,7141],{"className":7142,"style":2414},[2413],[1112,7144,7146],{"className":7145},[2418,2419,2420,2421],[1112,7147,4921],{"className":7148,"style":4972},[1155,1156,2421],[1112,7150,3693],{"className":7151},[3692],[1112,7153,7155],{"className":7154},[2402],[1112,7156,7158],{"className":7157,"style":5672},[2406],[1112,7159],{},[1112,7161,1251],{"className":7162},[1307],[1112,7164],{"className":7165,"style":1271},[1270],[1112,7167,3012],{"className":7168},[1275],[1112,7170],{"className":7171,"style":1271},[1270],[1112,7173,1203],{"className":7174,"style":1218},[1155,1156],[1112,7176,7177,7180],{"style":6912},[1112,7178],{"className":7179,"style":6903},[2413],[1112,7181,7183],{"className":7182},[1155],[1112,7184,7186],{"className":7185},[1155,2035],[1112,7187,12],{"className":7188},[1155,7066],[1112,7190,3693],{"className":7191},[3692],[1112,7193,7195],{"className":7194},[2402],[1112,7196,7198],{"className":7197,"style":6931},[2406],[1112,7199],{},[1112,7201],{"className":7202},[1307,7203],"nulldelimiter",[1091,7205,7206],{},"例如，下图为一个有向图和它的邻接矩阵:",[1091,7208,7209],{},[1634,7210],{"alt":2039,"src":7211},"https:\u002F\u002Fpic1.zhimg.com\u002F80\u002Fv2-5781f67fd4ba6ea0aec1bd16dbc0aa0a_720w.webp",[1091,7213,7214],{},"若 G 是网，则邻接矩阵可以定义为：",[1091,7216,7217],{},[1112,7218,7220,7321],{"className":7219},[1115],[1112,7221,7223],{"className":7222},[1119],[1121,7224,7225],{"xmlns":1123},[1125,7226,7227,7319],{},[1128,7228,7229,7231,7233,7235,7237,7239,7241,7243,7245],{},[1131,7230,6714],{},[1235,7232,6717],{"stretchy":1240},[1131,7234,4729],{},[1235,7236,6722],{"stretchy":1240},[1235,7238,6717],{"stretchy":1240},[1131,7240,4921],{},[1235,7242,6722],{"stretchy":1240},[1235,7244,1237],{},[1128,7246,7247,7249],{},[1235,7248,6735],{"fence":1142},[6737,7250,7251,7305],{"rowspacing":6739,"columnalign":6740,"columnspacing":6741},[6743,7252,7253,7259],{},[6746,7254,7255],{},[6749,7256,7257],{"scriptlevel":4734,"displaystyle":1240},[2763,7258,2765],{},[6746,7260,7261],{},[6749,7262,7263],{"scriptlevel":4734,"displaystyle":1240},[1128,7264,7265,7267,7273,7275,7281,7283,7285,7291,7293,7299,7301,7303],{},[2536,7266,1778],{},[3638,7268,7269,7271],{},[1131,7270,3001],{},[1131,7272,4729],{},[1235,7274,1246],{"separator":1142},[3638,7276,7277,7279],{},[1131,7278,3001],{},[1131,7280,4921],{},[2536,7282,6777],{},[1235,7284,1241],{"stretchy":1240},[3638,7286,7287,7289],{},[1131,7288,3001],{},[1131,7290,4729],{},[1235,7292,1246],{"separator":1142},[3638,7294,7295,7297],{},[1131,7296,3001],{},[1131,7298,4921],{},[1235,7300,1251],{"stretchy":1240},[1235,7302,3012],{},[1131,7304,1203],{},[6743,7306,7307,7313],{},[6746,7308,7309],{},[6749,7310,7311],{"scriptlevel":4734,"displaystyle":1240},[2763,7312,4734],{},[6746,7314,7315],{},[6749,7316,7317],{"scriptlevel":4734,"displaystyle":1240},[2536,7318,12],{},[1135,7320,6816],{"encoding":1137},[1112,7322,7324,7360],{"className":7323,"ariaHidden":1142},[1141],[1112,7325,7327,7330,7333,7336,7339,7342,7345,7348,7351,7354,7357],{"className":7326},[1146],[1112,7328],{"className":7329,"style":1285},[1150],[1112,7331,6714],{"className":7332},[1155,1156],[1112,7334,6717],{"className":7335},[1289],[1112,7337,4729],{"className":7338},[1155,1156],[1112,7340,6722],{"className":7341},[1307],[1112,7343,6717],{"className":7344},[1289],[1112,7346,4921],{"className":7347,"style":4972},[1155,1156],[1112,7349,6722],{"className":7350},[1307],[1112,7352],{"className":7353,"style":1271},[1270],[1112,7355,1237],{"className":7356},[1275],[1112,7358],{"className":7359,"style":1271},[1270],[1112,7361,7363,7366],{"className":7362},[1146],[1112,7364],{"className":7365,"style":6862},[1150],[1112,7367,7369,7375,7688],{"className":7368},[6866],[1112,7370,7372],{"className":7371,"style":6871},[1289,6870],[1112,7373,6735],{"className":7374},[6875,6876],[1112,7376,7378],{"className":7377},[1155],[1112,7379,7381,7426,7429],{"className":7380},[6737],[1112,7382,7384],{"className":7383},[6886],[1112,7385,7387,7418],{"className":7386},[2398,3669],[1112,7388,7390,7415],{"className":7389},[2402],[1112,7391,7393,7404],{"className":7392,"style":6896},[2406],[1112,7394,7395,7398],{"style":6899},[1112,7396],{"className":7397,"style":6903},[2413],[1112,7399,7401],{"className":7400},[1155],[1112,7402,2765],{"className":7403},[1155],[1112,7405,7406,7409],{"style":6912},[1112,7407],{"className":7408,"style":6903},[2413],[1112,7410,7412],{"className":7411},[1155],[1112,7413,4734],{"className":7414},[1155],[1112,7416,3693],{"className":7417},[3692],[1112,7419,7421],{"className":7420},[2402],[1112,7422,7424],{"className":7423,"style":6931},[2406],[1112,7425],{},[1112,7427],{"className":7428,"style":6938},[6937],[1112,7430,7432],{"className":7431},[6886],[1112,7433,7435,7680],{"className":7434},[2398,3669],[1112,7436,7438,7677],{"className":7437},[2402],[1112,7439,7441,7663],{"className":7440,"style":6896},[2406],[1112,7442,7443,7446],{"style":6899},[1112,7444],{"className":7445,"style":6903},[2413],[1112,7447,7449,7455,7495,7498,7501,7541,7547,7559,7562,7602,7605,7608,7648,7651,7654,7657,7660],{"className":7448},[1155],[1112,7450,7452],{"className":7451},[1155,2035],[1112,7453,1778],{"className":7454},[1155],[1112,7456,7458,7461],{"className":7457},[1155],[1112,7459,3001],{"className":7460,"style":1828},[1155,1156],[1112,7462,7464],{"className":7463},[2394],[1112,7465,7467,7487],{"className":7466},[2398,3669],[1112,7468,7470,7484],{"className":7469},[2402],[1112,7471,7473],{"className":7472,"style":5605},[2406],[1112,7474,7475,7478],{"style":3679},[1112,7476],{"className":7477,"style":2414},[2413],[1112,7479,7481],{"className":7480},[2418,2419,2420,2421],[1112,7482,4729],{"className":7483},[1155,1156,2421],[1112,7485,3693],{"className":7486},[3692],[1112,7488,7490],{"className":7489},[2402],[1112,7491,7493],{"className":7492,"style":3700},[2406],[1112,7494],{},[1112,7496,1246],{"className":7497},[1296],[1112,7499],{"className":7500,"style":1300},[1270],[1112,7502,7504,7507],{"className":7503},[1155],[1112,7505,3001],{"className":7506,"style":1828},[1155,1156],[1112,7508,7510],{"className":7509},[2394],[1112,7511,7513,7533],{"className":7512},[2398,3669],[1112,7514,7516,7530],{"className":7515},[2402],[1112,7517,7519],{"className":7518,"style":5605},[2406],[1112,7520,7521,7524],{"style":3679},[1112,7522],{"className":7523,"style":2414},[2413],[1112,7525,7527],{"className":7526},[2418,2419,2420,2421],[1112,7528,4921],{"className":7529,"style":4972},[1155,1156,2421],[1112,7531,3693],{"className":7532},[3692],[1112,7534,7536],{"className":7535},[2402],[1112,7537,7539],{"className":7538,"style":5672},[2406],[1112,7540],{},[1112,7542,7544],{"className":7543},[1155,2035],[1112,7545,1789],{"className":7546},[1155],[1112,7548,7550,7553,7556],{"className":7549},[1155,2035],[1112,7551,4048],{"className":7552},[1155],[1112,7554,7067],{"className":7555},[1155,7066],[1112,7557,4048],{"className":7558},[1155],[1112,7560,1241],{"className":7561},[1289],[1112,7563,7565,7568],{"className":7564},[1155],[1112,7566,3001],{"className":7567,"style":1828},[1155,1156],[1112,7569,7571],{"className":7570},[2394],[1112,7572,7574,7594],{"className":7573},[2398,3669],[1112,7575,7577,7591],{"className":7576},[2402],[1112,7578,7580],{"className":7579,"style":5605},[2406],[1112,7581,7582,7585],{"style":3679},[1112,7583],{"className":7584,"style":2414},[2413],[1112,7586,7588],{"className":7587},[2418,2419,2420,2421],[1112,7589,4729],{"className":7590},[1155,1156,2421],[1112,7592,3693],{"className":7593},[3692],[1112,7595,7597],{"className":7596},[2402],[1112,7598,7600],{"className":7599,"style":3700},[2406],[1112,7601],{},[1112,7603,1246],{"className":7604},[1296],[1112,7606],{"className":7607,"style":1300},[1270],[1112,7609,7611,7614],{"className":7610},[1155],[1112,7612,3001],{"className":7613,"style":1828},[1155,1156],[1112,7615,7617],{"className":7616},[2394],[1112,7618,7620,7640],{"className":7619},[2398,3669],[1112,7621,7623,7637],{"className":7622},[2402],[1112,7624,7626],{"className":7625,"style":5605},[2406],[1112,7627,7628,7631],{"style":3679},[1112,7629],{"className":7630,"style":2414},[2413],[1112,7632,7634],{"className":7633},[2418,2419,2420,2421],[1112,7635,4921],{"className":7636,"style":4972},[1155,1156,2421],[1112,7638,3693],{"className":7639},[3692],[1112,7641,7643],{"className":7642},[2402],[1112,7644,7646],{"className":7645,"style":5672},[2406],[1112,7647],{},[1112,7649,1251],{"className":7650},[1307],[1112,7652],{"className":7653,"style":1271},[1270],[1112,7655,3012],{"className":7656},[1275],[1112,7658],{"className":7659,"style":1271},[1270],[1112,7661,1203],{"className":7662,"style":1218},[1155,1156],[1112,7664,7665,7668],{"style":6912},[1112,7666],{"className":7667,"style":6903},[2413],[1112,7669,7671],{"className":7670},[1155],[1112,7672,7674],{"className":7673},[1155,2035],[1112,7675,12],{"className":7676},[1155,7066],[1112,7678,3693],{"className":7679},[3692],[1112,7681,7683],{"className":7682},[2402],[1112,7684,7686],{"className":7685,"style":6931},[2406],[1112,7687],{},[1112,7689],{"className":7690},[1307,7203],[1091,7692,7693,7694,7775],{},"其中，​",[1112,7695,7697,7720],{"className":7696},[1115],[1112,7698,7700],{"className":7699},[1119],[1121,7701,7702],{"xmlns":1123},[1125,7703,7704,7717],{},[1128,7705,7706],{},[3638,7707,7708,7711],{},[1131,7709,7710],{},"w",[1128,7712,7713,7715],{},[1131,7714,4729],{},[1131,7716,4921],{},[1135,7718,7719],{"encoding":1137},"w_{ij}",[1112,7721,7723],{"className":7722,"ariaHidden":1142},[1141],[1112,7724,7726,7729],{"className":7725},[1146],[1112,7727],{"className":7728,"style":5795},[1150],[1112,7730,7732,7736],{"className":7731},[1155],[1112,7733,7710],{"className":7734,"style":7735},[1155,1156],"margin-right:0.0269em;",[1112,7737,7739],{"className":7738},[2394],[1112,7740,7742,7767],{"className":7741},[2398,3669],[1112,7743,7745,7764],{"className":7744},[2402],[1112,7746,7748],{"className":7747,"style":5605},[2406],[1112,7749,7751,7754],{"style":7750},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[1112,7752],{"className":7753,"style":2414},[2413],[1112,7755,7757],{"className":7756},[2418,2419,2420,2421],[1112,7758,7760],{"className":7759},[1155,2421],[1112,7761,7763],{"className":7762,"style":4972},[1155,1156,2421],"ij",[1112,7765,3693],{"className":7766},[3692],[1112,7768,7770],{"className":7769},[2402],[1112,7771,7773],{"className":7772,"style":5672},[2406],[1112,7774],{},"  表示边上的权值，​ ∞  表示计算机允许的、大于所有边上权值的数。例如，下图为一个有向网和它的邻接矩阵:",[1091,7777,7778],{},[1634,7779],{"alt":2039,"src":7780},"https:\u002F\u002Fpic4.zhimg.com\u002F80\u002Fv2-362adaca006bc9620d2de9cc94b2abc7_720w.webp",[1091,7782,7783],{},"用邻接矩阵表示法表示图，除了一个用于存储邻接矩阵的二维数组外，还需要用一个一维数组来存储顶点信息。在 C 语言中，图的邻接矩阵类型描述如下：",[2030,7785,7790],{"className":7786,"code":7788,"language":7789,"meta":2039},[7787],"language-c","\u002F\u002F 表示极大值\n#define MaxInt 32767\n\u002F\u002F 最大顶点数\n#define MVNum 100\n​\n\u002F\u002F 假设顶点的数据类型为字符型\ntypedef char VerTexType;\n​\n\u002F\u002F 假设边的权值类型为整型\ntypedef int ArcType;\n​\ntypedef struct {\n    \u002F\u002F 顶点表\n    VerTexType vexs[MVNum];\n    \u002F\u002F 邻接矩阵\n    ArcType arcs[MVNum][MVNum];\n    \u002F\u002F 图的当前顶点数和边数\n    int vexnum, arcnum;\n}AMGraph;\n","c",[2037,7791,7788],{"__ignoreMap":2039},[6596,7793,7795],{"id":7794},"_312创建无向网",[1105,7796,7797],{},"3.1.2、创建无向网",[1091,7799,7800],{},"已知一个图的顶点和边，使用邻接矩阵表示法来创建此图的方法比较简单，下面以一个无向网为例来说明创建图的算法。该算法的步骤为：",[2041,7802,7803,7806,7809,7812],{},[2044,7804,7805],{},"输入总顶点数和总边数。",[2044,7807,7808],{},"依次输入顶点的信息存入顶点表中。",[2044,7810,7811],{},"初始化邻接矩阵，使每个权值初始化为极大值。",[2044,7813,7814],{},"构造邻接矩阵。依次输入每条边依附的顶点和其权值，确定两个顶点在图中的位置之后，使相应边赋予相应的权值，同时使其对称边赋予相同的权值。",[1091,7816,7817],{},"相应的算法描述为：",[2030,7819,7822],{"className":7820,"code":7821,"language":7789,"meta":2039},[7787],"\u002F\u002F 采用邻接矩阵表示法，创建无向网G\nStatus CreateUDN(AMGraph &G) {\n    \u002F\u002F 输入总顶点数和总边数\n    cin>>G.vexnum>>G.arcnum;\n​\n    \u002F\u002F 依次输入顶点信息\n    for(i=0; i\u003CG.vexnum; ++i)\n        cin>>G.vexs[i]\n​\n    \u002F\u002F 初始化邻接矩阵，边的权值均为极大值MaxInt\n    for(i=0; i\u003CG.vexnum; ++i)\n        for(j=0; j\u003CG.vexnum; ++j)\n            G.arcs[i][j]=MaxInt;\n​\n    \u002F\u002F 构造邻接矩阵\n    for(k=0;k\u003CG.arcnum;++k) {\n        \u002F\u002F 输入一条边依附的顶点和其权值\n        cin>v1>>v2>>w;\n        \u002F\u002F 确定v1和v2在G中的位置，即顶点数组的下标\n        i=LocateVex(G,v1);\n        j=LocateVex(G,v2);\n        \u002F\u002F 边\u003Cv1, v2>的权值置为w\n        G.arcs[i][j]=w;\n        \u002F\u002F 置\u003Cv1, v2>的对称边\u003Cv2, v1>的权值置为w\n        G.arcs[j][i]=G.arcs[i][j];\n    }\n​\n    return OK;\n}\n",[2037,7823,7821],{"__ignoreMap":2039},[1091,7825,7826,7827,4316],{},"该算法的时间复杂度是  ",[1112,7828,7830,7871],{"className":7829},[1115],[1112,7831,7833],{"className":7832},[1119],[1121,7834,7835],{"xmlns":1123},[1125,7836,7837,7868],{},[1128,7838,7839,7841,7843,7845,7847,7849,7851,7853,7855,7857,7859,7862,7864,7866],{},[1131,7840,3774],{},[1235,7842,1241],{"stretchy":1240},[1131,7844,4764],{},[1131,7846,6568],{},[1131,7848,1781],{},[1235,7850,1241],{"stretchy":1240},[1131,7852,2754],{},[2763,7854,2773],{},[1235,7856,1246],{"separator":1142},[1131,7858,2754],{},[1235,7860,7861],{},"∗",[1131,7863,2906],{},[1235,7865,1251],{"stretchy":1240},[1235,7867,1251],{"stretchy":1240},[1135,7869,7870],{"encoding":1137},"O(max(n2,n∗e))",[1112,7872,7874,7920],{"className":7873,"ariaHidden":1142},[1141],[1112,7875,7877,7880,7883,7886,7890,7893,7896,7899,7902,7905,7908,7911,7914,7917],{"className":7876},[1146],[1112,7878],{"className":7879,"style":1285},[1150],[1112,7881,3774],{"className":7882,"style":3614},[1155,1156],[1112,7884,1241],{"className":7885},[1289],[1112,7887,7889],{"className":7888},[1155,1156],"ma",[1112,7891,1781],{"className":7892},[1155,1156],[1112,7894,1241],{"className":7895},[1289],[1112,7897,2754],{"className":7898},[1155,1156],[1112,7900,2773],{"className":7901},[1155],[1112,7903,1246],{"className":7904},[1296],[1112,7906],{"className":7907,"style":1300},[1270],[1112,7909,2754],{"className":7910},[1155,1156],[1112,7912],{"className":7913,"style":1187},[1270],[1112,7915,7861],{"className":7916},[2801],[1112,7918],{"className":7919,"style":1187},[1270],[1112,7921,7923,7926,7929],{"className":7922},[1146],[1112,7924],{"className":7925,"style":1285},[1150],[1112,7927,2906],{"className":7928},[1155,1156],[1112,7930,7932],{"className":7931},[1307],"))",[1091,7934,7935],{},"若要建立无向图，只需对上述算法做两处小的改动：一是初始化邻接矩阵时，将边的权值均初始化为 0；二是构造邻接矩阵时，将权值 w 改为常量值 1 即可。同样，将该算法稍做修改即可建立一个有向网或有向图。",[6596,7937,7939],{"id":7938},"_313优缺点",[1105,7940,7941],{},"3.1.3、优缺点",[1091,7943,7944],{},"邻接矩阵表示法的优点是：",[2041,7946,7947,8032],{},[2044,7948,7949,7950,8031],{},"便于判断两个顶点之间是否有边，即根据 ",[1112,7951,7953,7983],{"className":7952},[1115],[1112,7954,7956],{"className":7955},[1119],[1121,7957,7958],{"xmlns":1123},[1125,7959,7960,7980],{},[1128,7961,7962,7964,7966,7968,7970,7972,7974,7976,7978],{},[1131,7963,6714],{},[1235,7965,6717],{"stretchy":1240},[1131,7967,4729],{},[1235,7969,6722],{"stretchy":1240},[1235,7971,6717],{"stretchy":1240},[1131,7973,4921],{},[1235,7975,6722],{"stretchy":1240},[1235,7977,1237],{},[2763,7979,4734],{},[1135,7981,7982],{"encoding":1137},"A[i][j]=0",[1112,7984,7986,8022],{"className":7985,"ariaHidden":1142},[1141],[1112,7987,7989,7992,7995,7998,8001,8004,8007,8010,8013,8016,8019],{"className":7988},[1146],[1112,7990],{"className":7991,"style":1285},[1150],[1112,7993,6714],{"className":7994},[1155,1156],[1112,7996,6717],{"className":7997},[1289],[1112,7999,4729],{"className":8000},[1155,1156],[1112,8002,6722],{"className":8003},[1307],[1112,8005,6717],{"className":8006},[1289],[1112,8008,4921],{"className":8009,"style":4972},[1155,1156],[1112,8011,6722],{"className":8012},[1307],[1112,8014],{"className":8015,"style":1271},[1270],[1112,8017,1237],{"className":8018},[1275],[1112,8020],{"className":8021,"style":1271},[1270],[1112,8023,8025,8028],{"className":8024},[1146],[1112,8026],{"className":8027,"style":4140},[1150],[1112,8029,4734],{"className":8030},[1155]," 或 1 来判断。",[2044,8033,8034],{},"便于计算各个顶点的度。对于无向图，邻接矩阵第 i 行元素之和就是顶点 i 的度；对于有向图，第 i 行元素之和就是顶点 i 的出度，第 i 列元素之和就是顶点 i 的入度。",[1091,8036,8037],{},"邻接矩阵表示法的缺点是：",[2041,8039,8040,8043,8046],{},[2044,8041,8042],{},"不便于增加和删除顶点。",[2044,8044,8045],{},"不便于统计边的数目，需要扫描邻接矩阵所有元素才能统计完毕，时间复杂度为 ​ O(n2) 。",[2044,8047,8048,8049,8122,8123,8172],{},"空间复杂度高。如果是有向图，n 个顶点需要 ​ n2  个单元存储边。如果是无向图，因其邻接矩阵是对称的，所以对规模较大的邻接矩阵可以采用压缩存储的方法，仅存储下三角（或上三角）的元素，这样需要 ",[1112,8050,8052,8080],{"className":8051},[1115],[1112,8053,8055],{"className":8054},[1119],[1121,8056,8057],{"xmlns":1123},[1125,8058,8059,8077],{},[1128,8060,8061,8063,8065,8067,8069,8071,8073,8075],{},[1131,8062,2754],{},[1235,8064,1241],{"stretchy":1240},[1131,8066,2754],{},[1235,8068,2761],{},[2763,8070,2765],{},[1235,8072,1251],{"stretchy":1240},[1131,8074,2770],{"mathvariant":2349},[2763,8076,2773],{},[1135,8078,8079],{"encoding":1137},"n(n−1)\u002F2",[1112,8081,8083,8107],{"className":8082,"ariaHidden":1142},[1141],[1112,8084,8086,8089,8092,8095,8098,8101,8104],{"className":8085},[1146],[1112,8087],{"className":8088,"style":1285},[1150],[1112,8090,2754],{"className":8091},[1155,1156],[1112,8093,1241],{"className":8094},[1289],[1112,8096,2754],{"className":8097},[1155,1156],[1112,8099],{"className":8100,"style":1187},[1270],[1112,8102,2761],{"className":8103},[2801],[1112,8105],{"className":8106,"style":1187},[1270],[1112,8108,8110,8113,8116,8119],{"className":8109},[1146],[1112,8111],{"className":8112,"style":1285},[1150],[1112,8114,2765],{"className":8115},[1155],[1112,8117,1251],{"className":8118},[1307],[1112,8120,2820],{"className":8121},[1155]," 个单元即可。但无论以何种方式存储，邻接矩阵表示法的空间复杂度均为  ",[1112,8124,8126,8148],{"className":8125},[1115],[1112,8127,8129],{"className":8128},[1119],[1121,8130,8131],{"xmlns":1123},[1125,8132,8133,8145],{},[1128,8134,8135,8137,8139,8141,8143],{},[1131,8136,3774],{},[1235,8138,1241],{"stretchy":1240},[1131,8140,2754],{},[2763,8142,2773],{},[1235,8144,1251],{"stretchy":1240},[1135,8146,8147],{"encoding":1137},"O(n2)",[1112,8149,8151],{"className":8150,"ariaHidden":1142},[1141],[1112,8152,8154,8157,8160,8163,8166,8169],{"className":8153},[1146],[1112,8155],{"className":8156,"style":1285},[1150],[1112,8158,3774],{"className":8159,"style":3614},[1155,1156],[1112,8161,1241],{"className":8162},[1289],[1112,8164,2754],{"className":8165},[1155,1156],[1112,8167,2773],{"className":8168},[1155],[1112,8170,1251],{"className":8171},[1307]," ​，这对于稀疏图而言尤其浪费空间。",[6596,8174,8176],{"id":8175},"_32邻接表",[1105,8177,8178],{},"3.2、邻接表",[6596,8180,8182],{"id":8181},"_321表示法",[1105,8183,8184],{},"3.2.1、表示法",[1091,8186,8187],{},"邻接表（Adjacency List）是图的一种链式存储结构。",[1091,8189,8190],{},"在邻接表中，对图中每个顶点 ​ 建立一个单链表，把与  vi ​ 相邻接的顶点放在这个链表中。邻接表中每个单链表的第一个结点存放有关顶点的信息，把这一结点看成链表的表头，其余结点存放有关边的信息，这样邻接表便由两部分组成：表头结点表和边表。",[1091,8192,8193],{},"表头结点表，由所有表头结点以顺序结构的形式存储，以便可以随机访问任一顶点的边链表。表头结点包括数据域（data）和链域（firstarc）两部分，如下图（a）所示。其中，数据域用于存储顶点 ​ vi  的名称或其他信息；链域用于指向链表中第一结点（即与顶点 ​ vi  相邻接的第一个邻接点）。",[1091,8195,8196],{},"边表，由表示图中顶点间关系的 2n 个边链表组成。边链表中边结点包括邻接点域、数据域和链域三部分，如下图（b）所示。其中，邻接点域指示与顶点  vi ​ 相邻接的点在图中的位置；数据域存储和边相关的信息，如权值等；链域指示与顶点 ​ vi  相邻接的下一个邻接点。",[1091,8198,8199],{},[1634,8200],{"alt":2039,"src":8201},"https:\u002F\u002Fpic2.zhimg.com\u002F80\u002Fv2-75524b79b1cc36f45383fee41b04856b_720w.webp",[1091,8203,8204],{},"例如，在下图（a）中有两个图，图（b）则是它们对应的邻接表。",[1091,8206,8207],{},[1634,8208],{"alt":2039,"src":8209},"https:\u002F\u002Fpicx.zhimg.com\u002F80\u002Fv2-1e2811444a3ec52f6d37a48b75272e45_720w.webp",[1091,8211,8212,8213,8282,8283,8352,8353,8422,8423,8492],{},"在无向图的邻接表中，顶点 ​ ",[1112,8214,8216,8233],{"className":8215},[1115],[1112,8217,8219],{"className":8218},[1119],[1121,8220,8221],{"xmlns":1123},[1125,8222,8223,8231],{},[1128,8224,8225],{},[3638,8226,8227,8229],{},[1131,8228,3001],{},[1131,8230,4729],{},[1135,8232,5714],{"encoding":1137},[1112,8234,8236],{"className":8235,"ariaHidden":1142},[1141],[1112,8237,8239,8242],{"className":8238},[1146],[1112,8240],{"className":8241,"style":3656},[1150],[1112,8243,8245,8248],{"className":8244},[1155],[1112,8246,3001],{"className":8247,"style":1828},[1155,1156],[1112,8249,8251],{"className":8250},[2394],[1112,8252,8254,8274],{"className":8253},[2398,3669],[1112,8255,8257,8271],{"className":8256},[2402],[1112,8258,8260],{"className":8259,"style":5605},[2406],[1112,8261,8262,8265],{"style":3679},[1112,8263],{"className":8264,"style":2414},[2413],[1112,8266,8268],{"className":8267},[2418,2419,2420,2421],[1112,8269,4729],{"className":8270},[1155,1156,2421],[1112,8272,3693],{"className":8273},[3692],[1112,8275,8277],{"className":8276},[2402],[1112,8278,8280],{"className":8279,"style":3700},[2406],[1112,8281],{},"  的度恰为第 i 个链表中的结点数；而在有向图中，第 i 个链表中的结点个数只是顶点 ​ 的出度，为求入度，必须遍历整个邻接表。在所有链表中，其邻接点域的值为 i 的结点的个数就是顶点   ",[1112,8284,8286,8303],{"className":8285},[1115],[1112,8287,8289],{"className":8288},[1119],[1121,8290,8291],{"xmlns":1123},[1125,8292,8293,8301],{},[1128,8294,8295],{},[3638,8296,8297,8299],{},[1131,8298,3001],{},[1131,8300,4729],{},[1135,8302,5714],{"encoding":1137},[1112,8304,8306],{"className":8305,"ariaHidden":1142},[1141],[1112,8307,8309,8312],{"className":8308},[1146],[1112,8310],{"className":8311,"style":3656},[1150],[1112,8313,8315,8318],{"className":8314},[1155],[1112,8316,3001],{"className":8317,"style":1828},[1155,1156],[1112,8319,8321],{"className":8320},[2394],[1112,8322,8324,8344],{"className":8323},[2398,3669],[1112,8325,8327,8341],{"className":8326},[2402],[1112,8328,8330],{"className":8329,"style":5605},[2406],[1112,8331,8332,8335],{"style":3679},[1112,8333],{"className":8334,"style":2414},[2413],[1112,8336,8338],{"className":8337},[2418,2419,2420,2421],[1112,8339,4729],{"className":8340},[1155,1156,2421],[1112,8342,3693],{"className":8343},[3692],[1112,8345,8347],{"className":8346},[2402],[1112,8348,8350],{"className":8349,"style":3700},[2406],[1112,8351],{}," ​ 的入度。有时，为了便于确定顶点的入度，可以建立一个有向图的逆邻接表，即对每个顶点 ​  ",[1112,8354,8356,8373],{"className":8355},[1115],[1112,8357,8359],{"className":8358},[1119],[1121,8360,8361],{"xmlns":1123},[1125,8362,8363,8371],{},[1128,8364,8365],{},[3638,8366,8367,8369],{},[1131,8368,3001],{},[1131,8370,4729],{},[1135,8372,5714],{"encoding":1137},[1112,8374,8376],{"className":8375,"ariaHidden":1142},[1141],[1112,8377,8379,8382],{"className":8378},[1146],[1112,8380],{"className":8381,"style":3656},[1150],[1112,8383,8385,8388],{"className":8384},[1155],[1112,8386,3001],{"className":8387,"style":1828},[1155,1156],[1112,8389,8391],{"className":8390},[2394],[1112,8392,8394,8414],{"className":8393},[2398,3669],[1112,8395,8397,8411],{"className":8396},[2402],[1112,8398,8400],{"className":8399,"style":5605},[2406],[1112,8401,8402,8405],{"style":3679},[1112,8403],{"className":8404,"style":2414},[2413],[1112,8406,8408],{"className":8407},[2418,2419,2420,2421],[1112,8409,4729],{"className":8410},[1155,1156,2421],[1112,8412,3693],{"className":8413},[3692],[1112,8415,8417],{"className":8416},[2402],[1112,8418,8420],{"className":8419,"style":3700},[2406],[1112,8421],{},"  建立一个链接所有进入 ​  ",[1112,8424,8426,8443],{"className":8425},[1115],[1112,8427,8429],{"className":8428},[1119],[1121,8430,8431],{"xmlns":1123},[1125,8432,8433,8441],{},[1128,8434,8435],{},[3638,8436,8437,8439],{},[1131,8438,3001],{},[1131,8440,4729],{},[1135,8442,5714],{"encoding":1137},[1112,8444,8446],{"className":8445,"ariaHidden":1142},[1141],[1112,8447,8449,8452],{"className":8448},[1146],[1112,8450],{"className":8451,"style":3656},[1150],[1112,8453,8455,8458],{"className":8454},[1155],[1112,8456,3001],{"className":8457,"style":1828},[1155,1156],[1112,8459,8461],{"className":8460},[2394],[1112,8462,8464,8484],{"className":8463},[2398,3669],[1112,8465,8467,8481],{"className":8466},[2402],[1112,8468,8470],{"className":8469,"style":5605},[2406],[1112,8471,8472,8475],{"style":3679},[1112,8473],{"className":8474,"style":2414},[2413],[1112,8476,8478],{"className":8477},[2418,2419,2420,2421],[1112,8479,4729],{"className":8480},[1155,1156,2421],[1112,8482,3693],{"className":8483},[3692],[1112,8485,8487],{"className":8486},[2402],[1112,8488,8490],{"className":8489,"style":3700},[2406],[1112,8491],{},"  的边的表。例如，下图（c）为有向图 ​ 的逆邻接表。",[1091,8494,8495],{},[1634,8496],{"alt":8497,"src":8498},"\n","https:\u002F\u002Fpicx.zhimg.com\u002F80\u002Fv2-6e24f163c4753af369826a7b4a2a6c43_720w.webp",[1091,8500,8501],{},"根据上述讨论，要定义一个邻接表，需要定义存放顶点的头结点和表示边的边结点。在 C 语言中，图的邻接表存储结构的类型描述如下：",[2030,8503,8506],{"className":8504,"code":8505,"language":7789,"meta":2039},[7787],"\u002F\u002F 最大顶点数\n#define MVNum 100\n​\n\u002F\u002F 边结点\ntypedef struct ArcNode {\n    \u002F\u002F 该边所指向的顶点的位置（邻接点域）\n    int adjvex;\n    \u002F\u002F 指向下一条边的指针（链域）\n    struct ArcNode *nextarc;\n    \u002F\u002F 和边相关的信息（数据域）\n    OtherInfo info;\n}ArcNode;\n​\n\u002F\u002F 顶点信息\ntypedef struct VNode {\n    VerTexType data;\n    \u002F\u002F 指向第一条依附该顶点的边的指针\n    ArcNode *firstarc;\n}VNode, AdjList[MVNum];\n​\n\u002F\u002F 邻接表\ntypedef struct {\n    AdjList vertices;\n    \u002F\u002F 图的当前顶点数和边数\n    int vexnum, arcnum;\n}ALGraph;\n",[2037,8507,8505],{"__ignoreMap":2039},[6596,8509,8511],{"id":8510},"_322创建无向图",[1105,8512,8513],{},"3.2.2、创建无向图",[1091,8515,8516],{},"基于邻接表表示法创建一个图，需要创建其相应的顶点表和边表。下面以一个无向图为例来说明创建图的算法。该算法步骤为：",[2041,8518,8519,8521,8524],{},[2044,8520,7805],{},[2044,8522,8523],{},"依次输入顶点的信息存入顶点表中，使每个表头结点的指针域初始化为 NULL。",[2044,8525,8526,8527,8596,8597,8666],{},"创建邻接表。依次输入每条边依附的两个顶点，确定这两个顶点的序号 i 和 j 之后，将此边结点分别插入 ​  ",[1112,8528,8530,8547],{"className":8529},[1115],[1112,8531,8533],{"className":8532},[1119],[1121,8534,8535],{"xmlns":1123},[1125,8536,8537,8545],{},[1128,8538,8539],{},[3638,8540,8541,8543],{},[1131,8542,3001],{},[1131,8544,4729],{},[1135,8546,5714],{"encoding":1137},[1112,8548,8550],{"className":8549,"ariaHidden":1142},[1141],[1112,8551,8553,8556],{"className":8552},[1146],[1112,8554],{"className":8555,"style":3656},[1150],[1112,8557,8559,8562],{"className":8558},[1155],[1112,8560,3001],{"className":8561,"style":1828},[1155,1156],[1112,8563,8565],{"className":8564},[2394],[1112,8566,8568,8588],{"className":8567},[2398,3669],[1112,8569,8571,8585],{"className":8570},[2402],[1112,8572,8574],{"className":8573,"style":5605},[2406],[1112,8575,8576,8579],{"style":3679},[1112,8577],{"className":8578,"style":2414},[2413],[1112,8580,8582],{"className":8581},[2418,2419,2420,2421],[1112,8583,4729],{"className":8584},[1155,1156,2421],[1112,8586,3693],{"className":8587},[3692],[1112,8589,8591],{"className":8590},[2402],[1112,8592,8594],{"className":8593,"style":3700},[2406],[1112,8595],{},"  和   ",[1112,8598,8600,8617],{"className":8599},[1115],[1112,8601,8603],{"className":8602},[1119],[1121,8604,8605],{"xmlns":1123},[1125,8606,8607,8615],{},[1128,8608,8609],{},[3638,8610,8611,8613],{},[1131,8612,3001],{},[1131,8614,4921],{},[1135,8616,5785],{"encoding":1137},[1112,8618,8620],{"className":8619,"ariaHidden":1142},[1141],[1112,8621,8623,8626],{"className":8622},[1146],[1112,8624],{"className":8625,"style":5795},[1150],[1112,8627,8629,8632],{"className":8628},[1155],[1112,8630,3001],{"className":8631,"style":1828},[1155,1156],[1112,8633,8635],{"className":8634},[2394],[1112,8636,8638,8658],{"className":8637},[2398,3669],[1112,8639,8641,8655],{"className":8640},[2402],[1112,8642,8644],{"className":8643,"style":5605},[2406],[1112,8645,8646,8649],{"style":3679},[1112,8647],{"className":8648,"style":2414},[2413],[1112,8650,8652],{"className":8651},[2418,2419,2420,2421],[1112,8653,4921],{"className":8654,"style":4972},[1155,1156,2421],[1112,8656,3693],{"className":8657},[3692],[1112,8659,8661],{"className":8660},[2402],[1112,8662,8664],{"className":8663,"style":5672},[2406],[1112,8665],{}," ​ 对应的两个边链表的头部。",[1091,8668,7817],{},[2030,8670,8673],{"className":8671,"code":8672,"language":7789,"meta":2039},[7787],"\u002F\u002F 采用邻接表表示法，创建无向图G\nStatus CreateUDG(ALGraph &G) {\n    \u002F\u002F 输入总顶点数，总边数\n    cin>>G.vexnum>>G.arcnum;\n​\n    \u002F\u002F 输入顶点信息，构造表头结点表\n    for(i=0; i\u003CG.vexnum; ++i) {\n        \u002F\u002F 输入顶点的值\n        cin>>G.vertices[i].data;\n        \u002F\u002F 初始化表头结点的指针域为NULL\n        G.vertices[i].firstarc=NULL;\n    }\n​\n    \u002F\u002F 输入各边，构造邻接表\n    for(k=0; k\u003CG.arcnum; ++k) {\n        \u002F\u002F 输入一条边依附的两个顶点\n        cin>>v1>>v2;\n\n        \u002F\u002F 确定v1和v2在G中的位置，即顶点在G.vertices中的下标\n        i=LocateVex(G,v1);\n        j=LocateVex(G,v2);\n​\n        \u002F\u002F 生成一个新的边结点*p1\n        p1=new ArcNode;\n        \u002F\u002F 邻接点序号为j\n        p2->adjvex=j;\n        \u002F\u002F 将新结点*p1插入到顶点vi的边表头部\n        p1->nextarc=G.vertices[i].firstarc;\n        G.vertices[i].firstarc=p1;\n​\n        \u002F\u002F 生成另一个对称的新边结点*p2\n        p2=new ArcNode;\n        \u002F\u002F 邻接点序号为i\n        p2->adjvex=i;\n        \u002F\u002F 将新结点*p2插入到顶点vj的边表头部\n        p2->nextarc=G.vertices[j].firstarc;\n        G.vertices[j].firstarc=p2;\n    }\n​\n    return OK;\n}\n",[2037,8674,8672],{"__ignoreMap":2039},[1091,8676,8677,8678,3945],{},"该算法的时间复杂度是 ",[1112,8679,8681,8705],{"className":8680},[1115],[1112,8682,8684],{"className":8683},[1119],[1121,8685,8686],{"xmlns":1123},[1125,8687,8688,8702],{},[1128,8689,8690,8692,8694,8696,8698,8700],{},[1131,8691,3774],{},[1235,8693,1241],{"stretchy":1240},[1131,8695,2754],{},[1235,8697,7861],{},[1131,8699,2906],{},[1235,8701,1251],{"stretchy":1240},[1135,8703,8704],{"encoding":1137},"O(n*e)",[1112,8706,8708,8732],{"className":8707,"ariaHidden":1142},[1141],[1112,8709,8711,8714,8717,8720,8723,8726,8729],{"className":8710},[1146],[1112,8712],{"className":8713,"style":1285},[1150],[1112,8715,3774],{"className":8716,"style":3614},[1155,1156],[1112,8718,1241],{"className":8719},[1289],[1112,8721,2754],{"className":8722},[1155,1156],[1112,8724],{"className":8725,"style":1187},[1270],[1112,8727,7861],{"className":8728},[2801],[1112,8730],{"className":8731,"style":1187},[1270],[1112,8733,8735,8738,8741],{"className":8734},[1146],[1112,8736],{"className":8737,"style":1285},[1150],[1112,8739,2906],{"className":8740},[1155,1156],[1112,8742,1251],{"className":8743},[1307],[1091,8745,8746,8747,8816],{},"建立有向图的邻接表与此类似，只是更加简单，每读入一个顶点对序号\u003Ci, j>，仅需生成一个邻接点序号为 j 的边表结点，并将其插入到 ​ ",[1112,8748,8750,8767],{"className":8749},[1115],[1112,8751,8753],{"className":8752},[1119],[1121,8754,8755],{"xmlns":1123},[1125,8756,8757,8765],{},[1128,8758,8759],{},[3638,8760,8761,8763],{},[1131,8762,3001],{},[1131,8764,4729],{},[1135,8766,5714],{"encoding":1137},[1112,8768,8770],{"className":8769,"ariaHidden":1142},[1141],[1112,8771,8773,8776],{"className":8772},[1146],[1112,8774],{"className":8775,"style":3656},[1150],[1112,8777,8779,8782],{"className":8778},[1155],[1112,8780,3001],{"className":8781,"style":1828},[1155,1156],[1112,8783,8785],{"className":8784},[2394],[1112,8786,8788,8808],{"className":8787},[2398,3669],[1112,8789,8791,8805],{"className":8790},[2402],[1112,8792,8794],{"className":8793,"style":5605},[2406],[1112,8795,8796,8799],{"style":3679},[1112,8797],{"className":8798,"style":2414},[2413],[1112,8800,8802],{"className":8801},[2418,2419,2420,2421],[1112,8803,4729],{"className":8804},[1155,1156,2421],[1112,8806,3693],{"className":8807},[3692],[1112,8809,8811],{"className":8810},[2402],[1112,8812,8814],{"className":8813,"style":3700},[2406],[1112,8815],{},"  的边链表头部即可。若要创建网的邻接表，可以将边的权值存储在 info 域中。",[1091,8818,8819],{},"需要注意的是，一个图的邻接矩阵表示是唯一的，但其邻接表表示不唯一，这是因为在邻接表表示中，各边表结点的链接次序取决于建立邻接表的算法，以及边的输入次序。",[6596,8821,8823],{"id":8822},"_323优缺点",[1105,8824,8825],{},"3.2.3、优缺点",[1091,8827,8828],{},"邻接矩阵和邻接表是图的两种最常用的存储结构，它们各有所长。与邻接矩阵相比，邻接表有其自己的优缺点。其优点是：",[2041,8830,8831,8834,8837],{},[2044,8832,8833],{},"便于增加和删除顶点。",[2044,8835,8836],{},"便于统计边的数目，按顶点表顺序扫描所有边表可得到边的数目，时间复杂度为 O(n+e)。",[2044,8838,8839],{},"空间效率高。对于一个具有 n 个顶点 e 条边的图 G，若 G 是无向图，则在其邻接表表示中有 n 个顶点表结点和 2e 个边表结点；若 G 是有向图，则在它的邻接表表示或逆邻接表表示中均有 n 个顶点表结点和 e 个边表结点。因此，邻接表或逆邻接表表示的空间复杂度为 O(n+e)，适合表示稀疏图。对于稠密图，考虑到邻接表中要附加链域，因此常采取邻接矩阵表示法。",[1091,8841,8842],{},"其缺点是：",[2041,8844,8845,8848],{},[2044,8846,8847],{},"不便于判断顶点之间是否有边，要判定 ​ 和 ​ 之间是否有边，就需扫描第 i 个边表，最坏情况下要耗费 O(n)时间。",[2044,8849,8850],{},"不便于计算有向图各个顶点的度。对于无向图，在邻接表表示中顶点 ​ 的度是第 i 个边表中的结点个数。在有向图的邻接表中，第 i 个边表上的结点个数是顶点 ​ vi  的出度，但求  vi ​ 的入度较困难，需遍历各顶点的边表。若有向图采用逆邻接表表示，则与邻接表表示相反，求顶点的入度容易，而求顶点的出度较难。",[6596,8852,8854],{"id":8853},"_33十字链表",[1105,8855,8856],{},"3.3、十字链表",[1091,8858,8859],{},"十字链表（Orthogonal List）是有向图的另一种链式存储结构。它可以看成是将有向图的邻接表和逆邻接表结合起来得到的一种链表。在十字链表中，对应于有向图中每一条弧有一个结点，对应于每个顶点也有一个结点。这些结点的结构形式如下图所示。",[1091,8861,8862],{},[1634,8863],{"alt":2039,"src":8864},"https:\u002F\u002Fpic4.zhimg.com\u002Fv2-4bb997c1c4861394a6576f934efaec75_1440w.jpg",[1091,8866,8867],{},"在弧结点中有 5 个域：其中尾域 tailvex 和头域 headvex 分别指示弧尾和弧头这两个顶点在图中的位置，链域 hlink 指向弧头相同的下一条弧，而链域 tlink 指向弧尾相同的下一条弧，info 域指向该弧的相关信息。弧头相同弧在同一链表上，弧尾相同的弧也在同一链表上。而它们的头结点即为顶点结点。",[1091,8869,8870],{},"顶点结点由 3 个域组成：其中 data 域存储和顶点相关的信息，如顶点的名称等；firstin 和 firstout 为两个链域，分别指向以该顶点为弧头或弧尾的第一个弧结点。",[1091,8872,8873],{},"例如，下图（b）是下图（a）所示图的十字链表。",[1091,8875,8876],{},[1634,8877],{"alt":2039,"src":8878},"https:\u002F\u002Fpic1.zhimg.com\u002F80\u002Fv2-b02a28cb3fe49e2c7a3875f31d014fd0_720w.webp",[1091,8880,8881],{},"若将有向图的邻接矩阵看成是稀疏矩阵的话，则十字链表也可以看成是邻接矩阵的链式存储结构，在图的十字链表中，弧结点所在的链表非循环链表，结点之间相对位置自然形成，不一定按顶点序号有序，表头结点即顶点结点，它们之间不是链接，而是顺序存储。",[1091,8883,8884],{},"在 C 语言中，有向图的十字链表存储结构的类型描述如下：",[2030,8886,8889],{"className":8887,"code":8888,"language":7789,"meta":2039},[7787],"#define MAX_VERTEX_NUM 20\n​\ntypedef struct ArcBox {\n    \u002F\u002F 该弧的尾和头结点的位置\n    int tailvex,headvex;\n    \u002F\u002F 分别为弧头相同和弧尾相同的弧的链域\n    struct ArcBox *hlink,*tlink;\n    \u002F\u002F 该弧相关信息的指针\n    InfoType *info;\n}ArcBox;\n​\ntypedef struct VexNode {\n    VertexType data;\n    \u002F\u002F 分别指向该顶点的第一条入弧和出弧\n    ArcBox *firstin, *firstout;\n}VexNode;\n​\ntypedef struct {\n    \u002F\u002F 表头向量\n    VexNode xlist[MAX_VERTEX_NUM];\n    \u002F\u002F 有向图的当前顶点数和弧数\n    int vexnum, arcnum;\n}QLGraph;\n",[2037,8890,8888],{"__ignoreMap":2039},[1091,8892,8893],{},"只要输入 n 个顶点的信息和 e 条弧的信息，便可建立该有向图的十字链表。建立十字链表的时间复杂度和建立邻接表是相同的。在十字链表中既容易找到以 ​ vi  为尾的弧，也容易找到以  vi ​ 为头的弧，因而容易求得顶点的出度和入度（或需要，可在建立十字链表的同时求出）。在某些有向图的应用中，十字链表是很有用的工具。",[6596,8895,8897],{"id":8896},"_34邻接多重表",[1105,8898,8899],{},"3.4、邻接多重表",[1091,8901,8902],{},"邻接多重表（Adjacency Multilist）是无向图的另一种链式存储结构。虽然邻接表是无向图的一种很有效的存储结构，在邻接表中容易求得顶点和边的各种信息。但是，在邻接表中每一条边  (vi,vj)  有 ​ 两个结点，分别在第 i 个和第 j 个链表中，这给某些图的操作带来不便。例如，在某些图的应用问题中，需要对边进行某种操作，如对已被搜索过的边作记号或删除一条边等，此时需要找到表示同一条边的两个结点。因此，在进行这一类操作的无向图的问题中采用邻接多重表作存储结构更为适宜。",[1091,8904,8905],{},[1634,8906],{"alt":2039,"src":8907},"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-21cf84e7eb759e2e0732d32827d79014_1440w.jpg",[1091,8909,8910],{},"邻接多重表的结构和十字链表类似。在邻接多重表中，每一条边用一个结点表示，它由 6 个域组成。其中，mark 为标志域，可用以标记该条边是否被搜索过；ivex 和 jvex 为该边依附的两个顶点在图中的位置；ilink 指向下一条依附于顶点 ivex 的边；jlink 指向下一条依附于顶点 jvex 的边；info 指向和边相关的各种信息的指针域。",[1091,8912,8913],{},"每一个顶点也用一个结点表示，它由两个域组成。其中，data 域存储和该顶点相关的信息，firstedge 域指示第一条依附于该顶点的边。",[1091,8915,8916],{},"例如，下图（b）是下图（a）所示图的邻接多重表。",[1091,8918,8919],{},[1634,8920],{"alt":2039,"src":8921},"https:\u002F\u002Fpic2.zhimg.com\u002F80\u002Fv2-157f84f916c008f3d847432673c79861_720w.webp",[1091,8923,8924],{},"在邻接多重表中，所有依附于同一顶点的边串联在同一链表中，由于每条边依附于两个顶点，则每个边结点同时链接在两个链表中。可见，对无向图而言，其邻接多重表和邻接表的差别，仅仅在于同一条边在邻接表中用两个结点表示，而在邻接多重表中只有一个结点。因此，除了在边结点中增加一个标志域外，邻接多重表所需的存储量和邻接表相同。",[1091,8926,8927],{},"在 C 语言中，邻接多重表的类型描述如下：",[2030,8929,8932],{"className":8930,"code":8931,"language":7789,"meta":2039},[7787],"#define MAX_VERTEX_NUM 20\n​\ntypedef enum{unvisited, visited} VisitIf;\n​\ntypedef struct EBox {\n    \u002F\u002F 访问标记\n    VisitIf mark;\n    \u002F\u002F 该边依附的两个顶点的位置\n    int ivex, jvex;\n    \u002F\u002F 分别指向依附这两个顶点的下一条边\n    struct EBox *ilink, *jlink;\n    \u002F\u002F 该边信息指针\n    InfoType *info;\n}Ebox;\n​\ntypedef struct VexBox {\n    VertexType data;\n    \u002F\u002F 指向第一条依附该顶点的边\n    EBox *firstedge;\n}VexBox;\n​\ntypedef struct {\n    VexBox adjmulist[MAX_VERTEX_NUM];\n    \u002F\u002F 无向图的当前顶点数和边数\n    int vexnum, edgenum;\n}AMLGraph;\n",[2037,8933,8931],{"__ignoreMap":2039},{"title":2039,"searchDepth":8935,"depth":8935,"links":8936},4,[8937,8939,8940],{"id":1103,"depth":8938,"text":1107},2,{"id":2244,"depth":8938,"text":2247},{"id":6560,"depth":8938,"text":6563,"children":8941},[8942,8944,8945,8946,8947,8948,8949,8950,8951,8952],{"id":6598,"depth":8943,"text":6601},3,{"id":6604,"depth":8943,"text":6607},{"id":7794,"depth":8943,"text":7797},{"id":7938,"depth":8943,"text":7941},{"id":8175,"depth":8943,"text":8178},{"id":8181,"depth":8943,"text":8184},{"id":8510,"depth":8943,"text":8513},{"id":8822,"depth":8943,"text":8825},{"id":8853,"depth":8943,"text":8856},{"id":8896,"depth":8943,"text":8899},"md",true,{"uuid":8956,"slots":8957},"8b815540-9d32-11f0-a5ba-d9d892ea74bd",{},18,{"title":1085,"description":1093},"posts\u002F数据结构\u002F2025-09-29-数据结构-图",[42,360],"lckkSVvmQi_oMEUC-Oed4h7CHHPfTGOXH9fOpiKKeQw",1790443288431]