1423:
27:
244:, a family of graphs derived from split graphs by doubling every vertex (so the clique comes to induce an antimatching and the independent set comes to induce a matching), figure prominently as one of five basic classes of perfect graphs from which all others can be formed in the proof by
658:
of an arbitrary graph measures the extent to which this inequality fails to be true. If a graph is not a split graph, then the smallest sequence of edge insertions and removals that make it into a split graph can be obtained by adding all missing edges between the
638:
263:, and vice versa. The split comparability graphs, and therefore also the split interval graphs, can be characterized in terms of a set of three forbidden induced subgraphs. The split
663:
vertices with the largest degrees, and removing all edges between pairs of the remaining vertices; the splittance counts the number of operations in this sequence.
717:
1572:
1124:
Proceedings of the Eighth
Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977)
19:
This article is about graphs that can be partitioned into a clique and an independent set. For cuts that form complete bipartite graphs, see
531:
1307:
1188:
1052:
1240:
Kézdy, André E.; Snevily, Hunter S.; Wang, Chi (1996), "Partitioning permutations into increasing and decreasing subsequences",
1242:
1145:
1340:
1066:
1015:
59:
1074:
1070:
249:
43:
312:
1399:
182:
755:
683:
210:
202:
1577:
1567:
1551:
1176:
320:
276:
453:
275:
are exactly the interval graphs that have interval graph complements; these are the permutation graphs of
762:
use the definition given here, which has been followed in the subsequent literature; for instance, it is
1206:
1140:
1119:
1079:
63:
55:
47:
1501:; Melnikow, O. I.; Kotov, V. M. (1981), "On graphs and degree sequences: the canonical decomposition",
1036:
93:
A split graph may have more than one partition into a clique and an independent set; for instance, the
20:
260:
1088:
214:
1498:
1453:
1419:
1394:
1184:
1048:
725:
691:
445:
272:
75:
323:
in a split graph. In any split graph, one of the following three possibilities must be true:
1487:
1349:
1316:
1252:
1220:
1154:
1098:
1062:
1024:
991:
687:
186:
83:
1535:
1514:
1479:. Translated as "Yet another method of enumerating unmarked combinatorial objects" (1990),
1475:
1445:
1412:
1386:
1330:
1297:
1266:
1232:
1198:
1168:
1131:
1110:
1005:
1531:
1510:
1471:
1441:
1408:
1382:
1326:
1293:
1262:
1228:
1194:
1164:
1127:
1106:
1001:
982:
Bender, E. A.; Richmond, L. B.; Wormald, N. C. (1985), "Almost all chordal graphs split",
751:
750:
had a more general definition, in which the graphs they called split graphs also included
469:
268:
51:
468:
One remarkable property of split graphs is that they can be recognized solely from their
1367:
1041:
675:
441:
304:
256:
217:
of subtrees of trees, split graphs are the intersection graphs of distinct substars of
1321:
1257:
1561:
1354:
1211:
1126:, Congressus Numerantium, vol. XIX, Winnipeg: Utilitas Math., pp. 311–315,
1028:
237:
206:
193:
on four or five vertices, or a pair of disjoint edges (the complement of a 4-cycle).
110:
is a split graph, the vertices of which can be partitioned in three different ways:
1363:
35:
758:
of bipartite graphs (that is, graphs that can be partitioned into two cliques).
457:
449:
437:
190:
1102:
996:
655:
280:
222:
218:
205:. Another characterization of split graphs involves complementation: they are
94:
1013:
Bertossi, Alan A. (1984), "Dominating sets for split and bipartite graphs",
1159:
319:. Thus, it is easy to identify the maximum clique, and complementarily the
1338:
Müller, Haiko (1996), "Hamiltonian
Circuits in Chordal Bipartite Graphs",
1424:"Canonical partition of a graph defined by the degrees of its vertices"
754:(that is, graphs that be partitioned into two independent sets) and the
710:
1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, ... (sequence
456:. It is also well known that the Minimum Dominating Set problem remains
26:
1491:
1224:
264:
1274:
McMorris, F. R.; Shier, D. R. (1983), "Representing chordal graphs on
1093:
1457:
633:{\displaystyle \sum _{i=1}^{m}d_{i}=m(m-1)+\sum _{i=m+1}^{n}d_{i}.}
1426:Каноническое разложение графа, определяемого степенями его вершин
245:
30:
A split graph, partitioned into a clique and an independent set.
1459:Еще один метод перечисления непомеченных комбинаторных объектов
690:. Using this fact, he determined a formula for the number of
1522:
Voss, H.-J. (1985), "Note on a paper of McMorris and Shier",
1047:, SIAM Monographs on Discrete Mathematics and Applications,
651:, and the remaining vertices constitute an independent set.
712:
647:
vertices with the largest degrees form a maximum clique in
444:, are similarly straightforward on split graphs. Finding a
201:
From the definition, split graphs are clearly closed under
956:
further investigates the degree sequences of split graphs.
225:
chordal graphs are split graphs; that is, in the limit as
213:
of which are also chordal. Just as chordal graphs are the
1481:
Mathematical notes of the
Academy of Sciences of the USSR
62:. Split graphs were first studied by Földes and
16:
Graph which partitions into a clique and independent set
941:
1209:; Simeone, Bruno (1981), "The splittance of a graph",
233:-vertex chordal graphs that are split approaches one.
874:, Corollary 7.1.1, p. 106, and Theorem 7.1.2, p. 107.
843:
534:
1143:(1977b), "Split graphs having Dilworth number two",
181:
Split graphs can be characterized in terms of their
1524:
1286:
949:
871:
831:
811:
763:
1040:
729:
632:
259:, then its complement is both a split graph and a
82:), where they called these graphs "polar graphs" (
79:
1550:A chapter on split graphs appears in the book by
1554:, "Algorithmic Graph Theory and Perfect Graphs".
883:
965:
933:
895:
855:
823:
803:
775:
759:
747:
71:
67:
8:
1077:(2006), "The strong perfect graph theorem",
408:is a maximal independent set. In this case,
295:be a split graph, partitioned into a clique
1181:Algorithmic Graph Theory and Perfect Graphs
937:
440:on more general graph families, including
436:Some other optimization problems that are
1434:Isv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk
1353:
1320:
1256:
1158:
1092:
995:
948:, Theorem 6.7 and Corollary 6.8, p. 154;
621:
611:
594:
560:
550:
539:
533:
1305:Merris, Russell (2003), "Split graphs",
1039:; Le, Van Bang; Spinrad, Jeremy (1999),
945:
922:
899:
859:
807:
791:
779:
255:If a graph is both a split graph and an
25:
942:Tyshkevich, Melnikow & Kotov (1981)
740:
1368:"Counting set covers and split graphs"
953:
911:
424:into a clique and an independent set,
844:Bender, Richmond & Wormald (1985)
671:
472:. Let the degree sequence of a graph
185:: a graph is split if and only if no
7:
827:
950:Brandstädt, Le & Spinrad (1999)
872:Brandstädt, Le & Spinrad (1999)
832:Brandstädt, Le & Spinrad (1999)
812:Brandstädt, Le & Spinrad (1999)
764:Brandstädt, Le & Spinrad (1999)
74:), and independently introduced by
728:result was also proved earlier by
229:goes to infinity, the fraction of
14:
1308:European Journal of Combinatorics
394:is a maximum independent set and
884:Kézdy, Snevily & Wang (1996)
730:Tyshkevich & Chernyak (1990)
525:is a split graph if and only if
452:even for split graphs which are
197:Relation to other graph families
1243:Journal of Combinatorial Theory
1146:Canadian Journal of Mathematics
432:is the maximum independent set.
383:is independent. In this case,
240:, so are the split graphs. The
1573:Intersection classes of graphs
1016:Information Processing Letters
698:vertices. For small values of
643:If this is the case, then the
584:
572:
1:
1322:10.1016/S0195-6698(03)00030-1
1258:10.1016/S0097-3165(96)80012-4
361:is a maximum independent set.
38:, a branch of mathematics, a
1375:Journal of Integer Sequences
1355:10.1016/0012-365x(95)00057-4
1029:10.1016/0020-0190(84)90126-1
682:-vertex split graphs are in
346:is complete. In this case,
250:Strong Perfect Graph Theorem
78: and Chernyak (
966:Hammer & Simeone (1981)
934:Hammer & Simeone (1981)
896:Hammer & Simeone (1981)
856:Földes & Hammer (1977b)
824:McMorris & Shier (1983)
804:Földes & Hammer (1977a)
776:Földes & Hammer (1977a)
760:Földes & Hammer (1977b)
748:Földes & Hammer (1977a)
428:is the maximum clique, and
307:in a split graph is either
236:Because chordal graphs are
183:forbidden induced subgraphs
1594:
1456:; Chernyak, A. A. (1990),
1422:; Chernyak, A. A. (1979),
1400:Doklady Akademii Nauk SSSR
1103:10.4007/annals.2006.164.51
952:, Theorem 13.3.2, p. 203.
18:
1122:(1977a), "Split graphs",
997:10.1017/S1446788700023077
766:, Definition 3.2.3, p.41.
684:one-to-one correspondence
87:
1458:
1425:
1177:Golumbic, Martin Charles
862:, Theorem 9.7, page 212.
503:be the largest value of
404:is a maximal clique and
357:is a maximum clique and
277:skew-merged permutations
246:Chudnovsky et al. (2006)
166:and the independent set
148:and the independent set
126:and the independent set
1552:Martin Charles Golumbic
1043:Graph Classes: A Survey
834:, Theorem 4.4.2, p. 52.
814:, Theorem 3.2.3, p. 41.
810:, Theorem 6.3, p. 151;
706:= 1, these numbers are
454:strongly chordal graphs
412:has a unique partition
321:maximum independent set
299:and an independent set
1207:Hammer, Peter Ladislaw
1160:10.4153/CJM-1977-069-1
1141:Hammer, Peter Ladislaw
1120:Hammer, Peter Ladislaw
984:J. Austral. Math. Soc.
902:, Theorem 6.2, p. 150.
794:, Theorem 6.1, p. 150.
782:, Theorem 6.3, p. 151.
634:
616:
555:
364:There exists a vertex
327:There exists a vertex
31:
1499:Tyshkevich, Regina I.
1454:Tyshkevich, Regina I.
1420:Tyshkevich, Regina I.
1395:Tyshkevich, Regina I.
1080:Annals of Mathematics
667:Counting split graphs
635:
590:
535:
29:
1341:Discrete Mathematics
532:
398:is a maximum clique.
287:Algorithmic problems
279:. Split graphs have
21:split (graph theory)
1037:Brandstädt, Andreas
261:comparability graph
242:double split graphs
215:intersection graphs
1492:10.1007/BF01240267
1225:10.1007/BF02579333
1183:, Academic Press,
1139:Földes, Stéphane;
1118:Földes, Stéphane;
630:
460:for split graphs.
281:cochromatic number
273:permutation graphs
32:
1063:Chudnovsky, Maria
938:Tyshkevich (1980)
446:Hamiltonian cycle
1585:
1538:
1517:
1486:(6): 1239–1245,
1478:
1448:
1431:
1415:
1389:
1372:
1364:Royle, Gordon F.
1358:
1357:
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795:
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773:
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752:bipartite graphs
745:
715:
702:, starting from
694:split graphs on
688:Sperner families
662:
650:
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639:
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631:
626:
625:
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610:
565:
564:
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524:
520:
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502:
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475:
470:degree sequences
464:Degree sequences
431:
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411:
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403:
397:
393:
382:
371:
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298:
294:
269:threshold graphs
267:are exactly the
187:induced subgraph
177:
165:
155:
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133:
125:
109:
89:
1593:
1592:
1588:
1587:
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1547:
1545:Further reading
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1521:
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1337:
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1117:
1067:Robertson, Neil
1061:
1055:
1035:
1012:
981:
977:
972:
964:
960:
946:Golumbic (1980)
932:
928:
923:Bertossi (1984)
921:
917:
910:
906:
900:Golumbic (1980)
894:
890:
882:
878:
870:
866:
860:Golumbic (1980)
854:
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822:
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808:Golumbic (1980)
802:
798:
792:Golumbic (1980)
790:
786:
780:Golumbic (1980)
774:
770:
746:
742:
738:
711:
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358:
347:
336:
332:
328:
316:
315:of a vertex in
311:itself, or the
308:
300:
296:
292:
289:
203:complementation
199:
167:
159:
149:
137:
127:
115:
97:
60:independent set
24:
17:
12:
11:
5:
1591:
1589:
1581:
1580:
1578:Perfect graphs
1575:
1570:
1568:Graph families
1560:
1559:
1556:
1555:
1546:
1543:
1541:
1540:
1519:
1505:(in Russian),
1495:
1466:(in Russian),
1450:
1436:(in Russian),
1416:
1403:(in Russian),
1391:
1360:
1335:
1315:(4): 413–430,
1302:
1278:
1271:
1251:(2): 353–359,
1237:
1219:(3): 275–284,
1203:
1189:
1173:
1153:(3): 666–672,
1136:
1115:
1059:
1053:
1033:
1010:
990:(2): 214–221,
978:
976:
973:
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442:graph coloring
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399:
362:
305:maximal clique
288:
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257:interval graph
207:chordal graphs
198:
195:
179:
178:
156:
134:
88:полярные графы
15:
13:
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6:
4:
3:
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1473:
1470:(6): 98–105,
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1214:
1213:
1212:Combinatorica
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1190:0-12-289260-7
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1087:(1): 51–229,
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1082:
1081:
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1075:Thomas, Robin
1072:
1071:Seymour, Paul
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1064:
1060:
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1054:0-89871-432-X
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912:Müller (1996)
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709:
708:
707:
705:
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693:
692:nonisomorphic
689:
686:with certain
685:
681:
677:
674:showed that (
673:
666:
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652:
627:
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326:
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314:
306:
303:. Then every
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49:
46:in which the
45:
41:
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28:
22:
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1523:
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1502:
1483:
1480:
1467:
1464:Mat. Zametki
1463:
1437:
1433:
1404:
1398:
1397:(1980), "",
1378:
1374:
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1094:math/0212070
1084:
1078:
1042:
1020:
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987:
983:
961:
929:
918:
907:
891:
879:
867:
851:
839:
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787:
771:
743:
723:
703:
699:
695:
679:
672:Royle (2000)
670:
653:
642:
516:
509:
492:
485:
478:
467:
435:
419:
415:
389:
385:
378:
374:
352:
348:
341:
337:
313:neighborhood
290:
271:. The split
254:
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39:
36:graph theory
33:
1530:: 319–322,
1503:Kibernetika
1407:: 677–679,
1348:: 291–298,
1292:: 489–494,
828:Voss (1985)
756:complements
726:enumerative
458:NP-complete
450:NP-complete
438:NP-complete
219:star graphs
211:complements
158:the clique
136:the clique
114:the clique
52:partitioned
40:split graph
1562:Categories
1245:, Series A
975:References
656:splittance
507:such that
499:, and let
372:such that
335:such that
223:Almost all
76:Tyshkevich
1440:: 14–26,
1023:: 37–40,
676:unlabeled
592:∑
579:−
537:∑
1366:(2000),
1179:(1980),
448:remains
265:cographs
48:vertices
1536:0803929
1515:0689420
1509:: 5–8,
1476:1102626
1446:0554162
1413:0587712
1387:1778996
1331:1975945
1298:0730144
1267:1370138
1233:0637832
1199:0562306
1169:0463041
1132:0505860
1111:2233847
1006:0770128
716:in the
713:A048194
521:. Then
248:of the
238:perfect
84:Russian
66: (
58:and an
54:into a
50:can be
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491:≥ … ≥
64:Hammer
56:clique
1430:(PDF)
1371:(PDF)
1089:arXiv
986:, A,
736:Notes
724:This
191:cycle
189:is a
72:1977b
68:1977a
44:graph
42:is a
1185:ISBN
1049:ISBN
718:OEIS
654:The
291:Let
209:the
95:path
80:1979
1488:doi
1350:doi
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340:∪ {
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