417:. This can be seen by building up the graph by adding pendant vertices, false twins, and true twins, at each step building up a corresponding set of chords representing the graph. Adding a pendant vertex corresponds to adding a chord near the endpoints of an existing chord so that it crosses only that chord; adding false twins corresponds to replacing a chord by two parallel chords crossing the same set of other chords; and adding true twins corresponds to replacing a chord by two chords that cross each other but are nearly parallel and cross the same set of other chords.
259:
20:
428:. Bipartite distance-hereditary graphs can be built up from a single vertex by adding only pendant vertices and false twins, since any true twin would form a triangle, but the pendant vertex and false twin operations preserve bipartiteness. Every bipartite distance-hereditary graph is
253:
They are the graphs that do not have as isometric subgraphs any cycle of length five or more, or any of three other graphs: a 5-cycle with one chord, a 5-cycle with two non-crossing chords, and a 6-cycle with a chord connecting opposite
324:
They are the graphs that have rank-width one, where the rank-width of a graph is defined as the minimum, over all hierarchical partitions of the vertices of the graph, of the maximum rank among certain submatrices of the graph's
731:
present a simple direct algorithm for maximum weighted independent sets in distance-hereditary graphs, based on parsing the graph into pendant vertices and twins, correcting a previous attempt at such an algorithm by
284:
Replace any vertex of the graph by a pair of vertices, each of which has as its neighbors the neighbors of the replaced vertex together with the other vertex of the pair. The new pair of vertices are called
147:
They are the graphs in which every induced path is a shortest path, or equivalently the graphs in which every non-shortest path has at least one edge connecting two non-consecutive path vertices.
478:
for more general classes of graphs. Thus, it is possible in polynomial time to find the maximum clique or maximum independent set in a distance-hereditary graph, or to find an optimal
352:
of five or more vertices), domino (six-vertex cycle plus a diagonal edge between two opposite vertices), or gem (five-vertex cycle plus two diagonals incident to the same vertex).
1128:
Gioan, Emeric; Paul, Christophe (2012), "Split decomposition and graph-labelled trees: Characterizations and fully dynamic algorithms for totally decomposable graphs",
455:
of any vertex in a distance-hereditary graph is a cograph. The transitive closure of the directed graph formed by choosing any set of orientations for the edges of any
459:
is distance-hereditary; the special case in which the tree is oriented consistently away from some vertex forms a subclass of distance-hereditary graphs known as the
277:
Replace any vertex of the graph by a pair of vertices, each of which has the same set of neighbors as the replaced vertex. The new pair of vertices are called
1266:
Hsieh, Sun-yuan; Ho, Chin-wen; Hsu, Tsan-sheng; Ko, Ming-tat (2002), "Efficient algorithms for the
Hamiltonian problem on distance-hereditary graphs",
1310:
1066:
1541:
505:
Several other optimization problems can also be solved more efficiently using algorithms specifically designed for distance-hereditary graphs. As
439:
The graphs that can be built from a single vertex by pendant vertices and true twins, without any false twin operations, are special cases of the
321:, forms two smaller graphs by replacing each of the two sides of the partition by a single vertex, and recursively partitions these two subgraphs.
1029:
1287:
267:
They are the graphs that can be built up from a single vertex by a sequence of the following three operations, as shown in the illustration:
1386:, SIAM Monographs on Discrete Mathematics and Applications, vol. 2, Philadelphia: Society for Industrial and Applied Mathematics,
377:, a graph in which every two induced paths between the same pair of vertices both have odd length or both have even length. Every even
737:
451:, which are therefore distance-hereditary; the cographs are exactly the disjoint unions of diameter-2 distance-hereditary graphs. The
1114:
Espelage, W.; Gurski, F.; Wanke, E. (2001), "How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time",
1416:
MĂĽller, Haiko; Nicolai, Falk (1993), "Polynomial time algorithms for
Hamiltonian problems on bipartite distance-hereditary graphs",
1399:
1316:
1271:
1119:
1072:
950:
946:
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867:
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Olaru and Sachs showed the α-perfection of the graphs in which every cycle of length five or more has a pair of crossing diagonals (
474:
Because distance-hereditary graphs are perfect, some optimization problems can be solved in polynomial time for them despite being
317:. In this decomposition, one finds a partition of the graph into two subsets, such that the edges separating the two subsets form a
908:
Cornelsen, Sabine; Di
Stefano, Gabriele (2005), "Treelike comparability graphs: characterization, recognition, and applications",
447:. The graphs that can be built from a single vertex by false twin and true twin operations, without any pendant vertices, are the
333:
471:
Distance-hereditary graphs can be recognized, and parsed into a sequence of pendant vertex and twin operations, in linear time.
1452:
820:
1418:
1351:
486:, they inherit polynomial time algorithms for circle graphs; for instance, it is possible determine in polynomial time the
1268:
Computing and
Combinatorics: 8th Annual International Conference, COCOON 2002 Singapore, August 15–17, 2002, Proceedings
1206:
1130:
69:
54:
are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph.
960:; Makowski, J. A.; Rotics, U. (2000), "Linear time solvable optimization problems on graphs on bounded clique width",
452:
989:
D'Atri, Alessandro; Moscarini, Marina (1988), "Distance-hereditary graphs, Steiner trees, and connected domination",
736:. Because distance-hereditary graphs are perfectly orderable, they can be optimally colored in linear time by using
991:
366:
510:
318:
299:
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with the maximum possible number of leaves) can be found in polynomial time on a distance-hereditary graph. A
1536:
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460:
429:
495:
1346:
1201:
295:
851:
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Combinatorial
Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969)
32:
1305:
Hui, Peter; Schaefer, Marcus; Ĺ tefankoviÄŤ, Daniel (2004), "Train tracks and confluent drawings", in
1065:; Meng, Jeremy Yu (2006), "Delta-confluent drawings", in Healy, Patrick; Nikolov, Nikola S. (eds.),
258:
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They are the graphs in which every cycle of length five or more has at least two crossing diagonals.
1062:
499:
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425:
314:
1157:
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1102:
1076:
1022:"A simple paradigm for graph recognition: application to cographs and distance hereditary graphs"
977:
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Cogis, O.; Thierry, E. (2005), "Computing maximum stable sets for distance-hereditary graphs",
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or
Hamiltonian path of any distance-hereditary graph can also be found in polynomial time.
19:
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143:
Distance-hereditary graphs can also be characterized in several other equivalent ways:
1042:
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of any circle graph and therefore of any distance-hereditary graph. Additionally, the
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It has been known for some time that the distance-hereditary graphs constitute an
1508:
891:
818:
Bandelt, Hans-JĂĽrgen; Mulder, Henry Martyn (1986), "Distance-hereditary graphs",
687:, Information System on Graph Classes and their Inclusions, retrieved 2016-09-30.
684:
1447:
921:
444:
378:
349:
1466:
1338:
1185:
1153:
1116:
Proc. 27th Int. Worksh. Graph-Theoretic
Concepts in Computer Science (WG 2001)
910:
Proc. 30th Int. Worksh. Graph-Theoretic
Concepts in Computer Science (WG 2004)
345:
1279:
1250:
1391:
1237:
Howorka, Edward (1977), "A characterization of distance-hereditary graphs",
487:
1172:; Rotics, Udi (2000), "On the clique-width of some perfect graph classes",
262:
Three operations by which any distance-hereditary graph can be constructed.
1483:
Sachs, Horst (1970), "On the Berge conjecture concerning perfect graphs",
973:
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482:
of any distance-hereditary graph. Because distance-hereditary graphs are
44:
1090:
494:
of any distance-hereditary graph is at most three. As a consequence, by
475:
448:
1081:
556:, α-perfection is an equivalent form of definition of perfect graphs.
1004:
1144:
257:
84:
The original definition of a distance-hereditary graph is a graph
18:
61:, although an equivalent class of graphs was already shown to be
1349:(1972), "Normal hypergraphs and the perfect graph conjecture",
274:
connected by a single edge to an existing vertex of the graph.
72:, but no intersection model was known until one was given by
1020:
Damiand, Guillaume; Habib, Michel; Paul, Christophe (2001),
862:, SIAM Monographs on Discrete Mathematics and Applications,
1204:; Maffray, Frédéric (1990), "Completely separable graphs",
391:
formed by connecting pairs of vertices at distance at most
294:
They are the graphs that can be completely decomposed into
57:
Distance-hereditary graphs were named and first studied by
409:
Every distance-hereditary graph can be represented as the
1514:
Information System on Graph
Classes and their Inclusions
1327:
International
Journal of Foundations of Computer Science
1174:
International Journal of Foundations of Computer Science
332:
They are the HHDG-free graphs, meaning that they have a
781:
612:
153:
They are the graphs in which, for every four vertices
1075:, vol. 3843, Springer-Verlag, pp. 165–176,
608:
502:
algorithms exist for many problems on these graphs.
1319:, vol. 3383, Springer-Verlag, pp. 318–328
1274:, vol. 2387, Springer-Verlag, pp. 51–75,
1122:, vol. 2204, Springer-Verlag, pp. 117–128
916:, vol. 3353, Springer-Verlag, pp. 46–57,
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611:and (for bipartite distance-hereditary graphs) by
719:but it was discovered to be erroneous by Damiand.
1325:Kloks, T. (1996), "Treewidth of circle graphs",
708:
740:to find a perfect ordering and then applying a
603:. A closely related decomposition was used for
16:Graph whose induced subgraphs preserve distance
769:
715:. An earlier linear time bound was claimed by
169:, at least two of the three sums of distances
1312:Proc. 12th Int. Symp. Graph Drawing (GD 2004)
1068:Proc. 13th Int. Symp. Graph Drawing (GD 2005)
801:
733:
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8:
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373:. Every distance-hereditary graph is also a
463:, which are also called chordal cographs.
1465:
1382:McKee, Terry A.; McMorris, F. R. (1999),
1364:
1219:
1143:
1080:
890:
833:
712:
600:
73:
1450:(2005), "Rank-width and vertex-minors",
854:; Le, Van Bang; Spinrad, Jeremy (1999),
1487:, Gordon and Breach, pp. 377–384,
782:Courcelle, Makowski & Rotics (2000)
636:
576:
529:
96:belong to a connected induced subgraph
58:
613:Hui, Schaefer & Ĺ tefankoviÄŤ (2004)
553:
753:
549:
361:Every distance-hereditary graph is a
7:
1239:The Quarterly Journal of Mathematics
685:Bipartite distance-hereditary graphs
609:Eppstein, Goodrich & Meng (2006)
1384:Topics in Intersection Graph Theory
786:Espelage, Gurski & Wanke (2001)
758:Brandstädt, Le & Spinrad (1999)
673:Brandstädt, Le & Spinrad (1999)
661:Brandstädt, Le & Spinrad (1999)
649:Brandstädt, Le & Spinrad (1999)
624:
589:Brandstädt, Le & Spinrad (1999)
14:
1317:Lecture Notes in Computer Science
1272:Lecture Notes in Computer Science
1120:Lecture Notes in Computer Science
1073:Lecture Notes in Computer Science
914:Lecture Notes in Computer Science
697:Cornelsen & Di Stefano (2005)
413:of chords on a circle, forming a
709:Damiand, Habib & Paul (2001)
357:Relation to other graph families
334:forbidden graph characterization
1453:Journal of Combinatorial Theory
821:Journal of Combinatorial Theory
420:A distance-hereditary graph is
381:of a distance-hereditary graph
136:is the same as the distance in
124:, so that the distance between
88:such that, if any two vertices
80:Definition and characterization
1542:Intersection classes of graphs
1419:Information Processing Letters
675:, Theorem 10.6.14, p.164.
591:, Theorem 10.1.1, p. 147.
1:
1043:10.1016/S0304-3975(00)00234-6
507:D'Atri & Moscarini (1988)
1509:"Distance-hereditary graphs"
1432:10.1016/0020-0190(93)90100-N
1366:10.1016/0012-365X(72)90006-4
1221:10.1016/0166-218X(90)90131-U
1207:Discrete Applied Mathematics
1131:Discrete Applied Mathematics
1030:Theoretical Computer Science
892:10.1016/j.disopt.2005.03.004
835:10.1016/0095-8956(86)90043-2
770:Golumbic & Rotics (2000)
329:determined by the partition.
70:intersection class of graphs
65:in 1970 by Olaru and Sachs.
23:A distance-hereditary graph.
962:Theory of Computing Systems
922:10.1007/978-3-540-30559-0_4
802:MĂĽller & Nicolai (1993)
734:Hammer & Maffray (1990)
717:Hammer & Maffray (1990)
585:Hammer & Maffray (1990)
581:Bandelt & Mulder (1986)
566:McKee & McMorris (1999)
537:Hammer & Maffray (1990)
319:complete bipartite subgraph
1558:
1467:10.1016/j.jctb.2005.03.003
729:Cogis & Thierry (2005)
43:) is a graph in which the
41:completely separable graph
1339:10.1142/S0129054196000099
1186:10.1142/S0129054100000260
1154:10.1016/j.dam.2011.05.007
992:SIAM Journal on Computing
367:perfectly orderable graph
300:complete bipartite graphs
37:distance-hereditary graph
1280:10.1007/3-540-45655-4_10
1170:Golumbic, Martin Charles
511:connected dominating set
461:trivially perfect graphs
250:are equal to each other.
1392:10.1137/1.9780898719802
858:Graph Classes: A Survey
713:Gioan & Paul (2012)
601:Gioan & Paul (2012)
74:Gioan & Paul (2012)
1251:10.1093/qmath/28.4.417
1202:Hammer, Peter Ladislaw
552:, Theorem 5). By
365:, more specifically a
336:according to which no
263:
120:must be a subgraph of
24:
974:10.1007/s002249910009
879:Discrete Optimization
424:if and only if it is
261:
22:
1352:Discrete Mathematics
1063:Goodrich, Michael T.
385:(that is, the graph
340:can be a house (the
33:discrete mathematics
1091:10.1007/11618058_16
852:Brandstädt, Andreas
798:Hsieh et al. (2002)
651:, pp. 70–71 and 82.
513:(or equivalently a
500:dynamic programming
496:Courcelle's theorem
315:split decomposition
411:intersection graph
264:
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519:Hamiltonian cycle
430:chordal bipartite
344:of a five-vertex
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108:connecting
1526:Categories
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744:algorithm.
625:Oum (2005)
550:Sachs 1970
467:Algorithms
346:path graph
287:true twins
270:Add a new
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760:, p. 170.
488:treewidth
422:bipartite
254:vertices.
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