Clique-Width of Graph Classes Defined by Two Forbidden Induced Subgraphs

Abstract

If a graph has no induced subgraph isomorphic to any graph in a finite family {H1,. .. , Hp}, it is said to be (H1,. .. , Hp)-free. The class of H-free graphs has bounded clique-width if and only if H is an induced subgraph of the 4-vertex path P4. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs H1 and H2. Prior to our study it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases we determine new graph classes of bounded clique-width and new graph classes of unbounded clique-width. For obtaining the latter results we first present a new, generic construction for graph classes of unbounded clique-width. Our results settle the boundedness or unboundedness of the clique-width of the class of (H1, H2)-free graphs (i) for all pairs (H1, H2), both of which are connected, except two nonequivalent cases, and (ii) for all pairs (H1, H2), at least one of which is not connected, except 11 non-equivalent cases. We also consider classes characterized by forbidding a finite family of graphs {H1,. .. , Hp} as subgraphs, minors and topological minors, respectively, and completely determine which of these classes have bounded clique-width. Finally, we show algorithmic consequences of our results for the graph colouring problem restricted to (H1, H2)-free graphs.

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.…”

Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes

“…For monogenic classes, the conjecture is true. In this case, the two notions even coincide: a class of graphs defined by a single forbidden induced subgraph H is well-quasi-ordered if and only if it has bounded clique-width if and only if H is an induced subgraph of P 4 (see, for instance, [14,16,21]). …”

Bounding the clique-width of H-free split graphs

“…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).…”

“…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). Similar studies have been performed for variants of clique-width, such as linear clique-width [26] and power-bounded clique-width [2].…”