Bounding the Clique‐Width of H‐Free Chordal Graphs

Abstract

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 42This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. BOUNDING CLIQUE-WIDTH OF H-FREE CHORDAL GRAPHS 43of 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.

Bounding the clique-width of H-free split graphs

“…However, the proof of Theorem 2 in [3] relies on results from this paper. We therefore prove all the results in this paper without relying on Theorem 2 or any other results from [3].…”

“…Both of these graphs have seven vertices. The six-vertex induced subgraphs of F 4 are: bull +P 1 , F 1 , F 3 and 3 and Q. These graphs and their complements are precisely the cases listed in Theorem 4 (and for which we prove boundedness in Section 3).…”

“…A graph class is said to be of bounded clique-width if there is a constant c such that the clique-width of every graph in the class is at most c. Much research has been done identifying whether or not various classes have bounded clique-width [1][2][3][5][6][7][8][9][10][11][15][16][17][18][19]23,[29][30][31][32]. For instance, the Information System on Graph Classes and their Inclusions [20] maintains a record of graph classes for which this is known.…”

“…For instance, the Information System on Graph Classes and their Inclusions [20] maintains a record of graph classes for which this is known. In a recent series of papers [3,16,19] the clique-width of graph classes characterized by two forbidden induced subgraphs was investigated. In particular we refer to [19] for details on how new results can be combined with known results to give a classification for all but 13 open cases (up to an equivalence relation).…”

Classifying the clique-width of H-free bipartite graphs

“…As mentioned in Section 1, many results exist in the literature. In a series of follow-up papers [3,4,16,18] we have tried to address this question by determining classes of (H 1 , H 2 )-free (general) graphs, H-free split graphs, H-free chordal graphs and H-free weakly chordal graphs of bounded and unbounded clique-width. In each of these papers, we have applied our results for H-free bipartite graphs as useful lemmas.…”