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.

The class of H-free graphs has bounded clique-width if and only if H is an induced subgraph of the 4-vertex path P 4 . We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs H 1 and H 2 . 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 (H 1 , H 2 )-free graphs (i) for all pairs (H 1 , H 2 ), both of which are connected, except two non-equivalent cases, and (ii) for all pairs (H 1 , H 2 ), 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 {H 1 , . . . , H p } 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 (H 1 , H 2 )-free graphs.

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 = ∅.

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let G be a bipartite graph, and let H be a bipartite graph with a fixed bipartition (BH , WH ). We consider three different, natural ways of forbidding H as an induced subgraph in G.

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.

Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Theoretical computer science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version was subsequently published in Theoretical computer science, 522, 2014, 10.1016/j.tcs.2013.12.004 Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The Colouring problem is that of testing whether a given graph has a k-colouring for some given integer k. If a graph contains no induced subgraph isomorphic to any graph in some family H, then it is called H-free. The complexity of Colouring for H-free graphs with |H| = 1 has been completely classified. When |H| = 2, the classification is still wide open, although many partial results are known. We continue this line of research and forbid induced subgraphs {H1, H2}, where we allow H1 to have a single edge and H2 to have a single nonedge. Instead of showing only polynomial-time solvability, we prove that Colouring on such graphs is fixed-parameter tractable when parameterized by |H1| + |H2|. As a by-product, we obtain the same result both for the problem of determining a maximum independent set and for the problem of determining a maximum clique.

Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Theoretical computer science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version was subsequently published in Theoretical computer science, 522, 2014, 10.1016/j.tcs.2013.12.004 Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The Colouring problem is that of testing whether a given graph has a k-colouring for some given integer k. If a graph contains no induced subgraph isomorphic to any graph in some family H, then it is called H-free. The complexity of Colouring for H-free graphs with |H| = 1 has been completely classified. When |H| = 2, the classification is still wide open, although many partial results are known. We continue this line of research and forbid induced subgraphs {H1, H2}, where we allow H1 to have a single edge and H2 to have a single nonedge. Instead of showing only polynomial-time solvability, we prove that Colouring on such graphs is fixed-parameter tractable when parameterized by |H1| + |H2|. As a by-product, we obtain the same result both for the problem of determining a maximum independent set and for the problem of determining a maximum clique.