Bounding the Clique-Width of H-free Chordal Graphs

Abstract

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 = ∅.

Square-Free Graphs with No Six-Vertex Induced Path

“…Then by (2), a + b + t = q, and by using the q-cliques given by (2), we deduce successively that |Q x | = a, |Q v1 | = t and |Q v2 | = b. Then again by (2) and by our assumption, since Q xv2v3 and Q xyv3 are q-cliques, we see that |Q v3 | = b = t. So, q = a + 2t. Since Q yv3v4 is a q-clique (by (2)), we have |Q v4 | = a.…”

“…Likewise, one of Q zv5v6 and Q xv2v3 is a q-clique, and one of Q xv1v2 and Q yv4v5 is a q-clique. Up to symmetry this yields the possibilities described in (2). Thus we may assume that (2) holds.…”

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

“… for the class of ‐free graphs. 11.The class of ‐free graphs has bounded clique width , hence we apply Theorem (i). 12.This was proved by Malyshev for and by Malyshev for .…”

Bounding the Clique‐Width of H‐Free Chordal Graphs

“…We thank the two anonymous reviewers for their very detailed comments, which helped to improve the presentation of the paper. An extended abstract of this article appeared in the proceedings of MFCS 2015 . The research in this article was supported by EPSRC (EP/K025090/1).…”