Classifying the clique-width of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si106.gif" display="inline" overflow="scroll"><mml:mi>H</mml:mi></mml:math>-free bipartite graphs

Electronic Notes in Discrete Mathematics

Huang

et al. 2015

Self Cite

A graph is H-free if it has no induced subgraph isomorphic to H. We continue a study into the boundedness of clique-width of subclasses of perfect graphs. We identify five new classes of H-free split graphs whose clique-width is bounded. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H-free split graphs has bounded cliquewidth.

Section: Theorem 2 ([3])mentioning

confidence: 98%

Section: Theorem 3 ([19]) Let H Be a Graph The Class Of H-free Bipamentioning

confidence: 99%

See 1 more Smart Citation

Bounding the Clique-Width of H -free Split Graphs

Brandstädt

Electronic Notes in Discrete Mathematics

Huang

et al. 2015

Self Cite

“…We will use the following characterization of graphs H for which the class of Hfree bipartite graphs has bounded clique-width (which is similar to a characterization of Lozin and Volz [43] for a different variant of the notion of H-freeness in bipartite graphs, see [24] for an explanation of the difference).…”

Section: Lemma 13 ([23]) Let H Be a Graph The Class Of H-free Graphmentioning

confidence: 99%

Bounding the Clique-Width of H-free Chordal Graphs

Brandstädt

Mathematical Foundations of Computer Science 2015

Huang

et al. 2015

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A graph is H-free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le, and Mosca erroneously claimed that the gem and co-gem are the only two 1-vertex P 4 -extensions H for which the class of H-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we find four new classes of H-free chordal graphs of bounded clique-width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H-free chordal graphs has bounded clique-width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of (2P 1 + P 3 , K 4 )-free graphs has bounded clique-width via a reduction to K 4 -free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique-width of H-free weakly chordal graphs.1 See also Information System on Graph Classes and their Inclusions [30], which keeps a record of graph classes for which (un)boundedness of clique-width is known. 2 This follows from results [22,33,45,57] that assume the existence of a so-called c-expression of the input graph G ∈ G combined with a result [55] that such a cexpression can be obtained in cubic time for some c ≤ 8 cw(G) − 1, where cw(G) is the clique-width of the graph G. Lemma 3 ([6]). If a prime graph G contains an induced 2P 1 + P 2 , then it contains an induced P 1 + P 4 , d-A or d-domino (see Fig. 4). Lemma 4 ([15]). If a prime graph G contains an induced subgraph isomorphic to P 1 + P 4 , then it contains one of the graphs in Figure 5 as an induced subgraph.We also use the following structural lemma due to Olariu.Lemma 5 ([54]). Every prime (bull, house)-free graph (see Fig. 6) is either K 3 -free or the complement of a 2P 2 -free bipartite graph.Lemma 32. If a prime (2P 2 , C 5 , S 1,1,2 )-free graph G contains an induced subgraph isomorphic to the net (see Fig. 5), then G is a thin spider.Proof. Suppose that G is a prime (2P 2 , C 5 , S 1,1,2 )-free graph and suppose that G contains a net, say N, with vertices a 1 , a 2 , a 3 -vertices of N), and the only edges between a 1 , a 2 , a 3 and bWe partition M as follows: For i ∈ {1, . . . , 5}, let M i be the set of vertices in M with exactly i neighbours in V (N). Let U be the set of vertices in M adjacent to every vertex of V (N). Let Z be the set of vertices in M with no neighbours in V (N). Note that Z is an independent set in G, since G is 2P 2 -free.We now analyze the structure of G through a series of claims.Claim 1. M 1 ∪ M 2 ∪ M 5 = ∅.

“…Then, apply join operations η 11,22 , η 12,23 and η 13,21 . Finally, apply the relabelling operations ρ 21→11 , ρ 22→12 and ρ 23→13 .…”

Section: The Decompositionmentioning

confidence: 99%

Clique-Width and Well-Quasi-Ordering of Triangle-Free Graph Classes

Graph-Theoretic Concepts in Computer Science

Paulusma

2017

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Abstract. Daligault, Rao and Thomassé asked whether every hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev (JCTB 2017+) gave a negative answer to this question, but their counterexample is a class that can only be characterised by infinitely many forbidden induced subgraphs. This raises the issue of whether the question has a positive answer for finitely defined hereditary graph classes. Apart from two stubborn cases, this has been confirmed when at most two induced subgraphs H1, H2 are forbidden. We confirm it for one of the two stubborn cases, namely for the (H1, H2) = (triangle, P2 + P4) case, by proving that the class of (triangle, P2 + P4)-free graphs has bounded clique-width and is well-quasi-ordered. Our technique is based on a special decomposition of 3-partite graphs. We also use this technique to prove that the class of (triangle, P1 + P5)-free graphs, which is known to have bounded cliquewidth, is well-quasi-ordered. Our results enable us to complete the classification of graphs H for which the class of (triangle, H)-free graphs is well-quasi-ordered.