Clique-width of Graph Classes Defined byTwo Forbidden Induced SubgraphsThe research in this paper was supported by EPSRC (EP/G043434/1 and EP/K025090/1) and ANR (TODO ANR-09-EMER-010).

Clique-width of Graph Classes Defined by
Two Forbidden Induced Subgraphsthanks: The research in this paper was supported by EPSRC (EP/G043434/1 and EP/K025090/1) and ANR (TODO ANR-09-EMER-010).

Konrad K. Dabrowski School of Engineering and Computing Sciences, Durham University,
Science Laboratories, South Road,
Durham DH1 3LE, United Kingdom
{konrad.dabrowski,daniel.paulusma}@durham.ac.uk
   Daniël Paulusma School of Engineering and Computing Sciences, Durham University,
Science Laboratories, South Road,
Durham DH1 3LE, United Kingdom
{konrad.dabrowski,daniel.paulusma}@durham.ac.uk
Abstract

If a graph has no induced subgraph isomorphic to any graph in a finite family , it is said to be -free. The class of -free graphs has bounded clique-width if and only if is an induced subgraph of the 4-vertex path . We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs  and . 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 -free graphs

  1. for all pairs , both of which are connected, except two non-equivalent cases, and

  2. for all pairs , 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 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 -free graphs.

Keywords:
clique-width, forbidden induced subgraph, graph class

1 Introduction

Clique-width is a well-known graph parameter studied both in a structural and in an algorithmic context; we refer to the surveys of Gurski [30] and Kamiński, Lozin and Milanič [34] for an in-depth study of the properties of clique-width. However, our understanding of clique-width, which is one of the most difficult graph parameters to deal with, is still very limited. For example, no polynomial-time algorithms are known for computing the clique-width of very restricted graph classes, such as unit interval graphs, or for deciding whether a graph has clique-width at most for any fixed (as an aside, we note that such an algorithm does exist for  [13]).

In order to get more structural insight into clique-width, we are interested in determining whether the clique-width of some given class of graphs is bounded, that is, whether there exists a constant  such that every graph from the class has clique-width at most  (our secondary motivation is algorithmic, as we will explain in detail later). The graph classes that we consider consist of graphs in which one or more specified graphs are forbidden as a “pattern”. In particular, we consider classes of graphs that contain no graph from some specified family as an induced subgraph; such classes are said to be -free. Our research is well embedded in the literature, as there are many papers that determine the boundedness or unboundedness of the clique-width of graph classes characterized by one or more forbidden induced subgraphs; see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 21, 22, 29, 38, 39, 40, 41].

As we show later, it is not difficult to verify that the class of -free graphs has bounded clique-width if and only if is an induced subgraph of the 4-vertex path . Hence, it is natural to consider the following problem:

For which pairs does the class of -free graphs have bounded clique-width?

In this paper we address this question by narrowing the gap between the known and open cases significantly; in particular we show that the number of open cases is finite. We emphasise that the underlying research question is: what kind of properties of a graph class ensure that its clique-width is bounded? Our paper is to be interpreted as a further step towards this direction, and in our research project (see also [3, 20, 22]) we aim to develop general techniques for attacking a number of the open cases simultaneously.

Algorithmic Motivation

For problems that are NP-complete in general, one naturally seeks to find subclasses of graphs on which they are tractable, and graph classes of bounded clique-width have been studied extensively for this purpose, as we discuss below.

Courcelle, Makowsky and Rotics [17] showed that all MSO graph problems, which are problems definable in Monadic Second Order Logic using quantifiers on vertices but not on edges, can be solved in linear time on graphs with clique-width at most , provided that a -expression of the input graph is given. Later, Espelage, Gurski and Wanke [25], Kobler and Rotics [35] and Rao [50] proved the same result for many non-MSO graph problems. Although computing the clique-width of a given graph is NP-hard, as shown by Fellows, Rosamond, Rotics and Szeider [26], it is possible to find an -expression for any -vertex graph with clique-width at most in cubic time. This is a result of Oum [45] after a similar result (with a worse bound and running time) had already been shown by Oum and Seymour [46]. Hence, the NP-complete problems considered in the aforementioned papers [17, 25, 35, 50] are all polynomial-time solvable on any graph class of bounded clique-width even if no -expression of the input graph is given.

As a consequence of the above, when solving an NP-complete problem on some graph class , it is natural to try to determine first whether the clique-width of is bounded. In particular this is the case if we aim to determine the computational complexity of some NP-complete problem when restricted to graph classes characterized by some common type of property. This property may be the absence of a family of forbidden induced subgraphs and we may want to classify for which families of graphs the problem is still NP-hard and for which ones it becomes polynomial-time solvable (in order to increase our understanding of the hardness of the problem in general). We give examples later.

Our Results

In Section 2 we state a number of basic results on clique-width and two results on -free bipartite graphs that we showed in a very recent paper [22]; we need these results for proving our new results. We then identify a number of new classes of -free graphs of bounded clique-width (Section 3) and unbounded clique-width (Section 4). In particular, the new unbounded cases are obtained from a new, general construction for graph classes of unbounded clique-width. In Section 5, we first observe for which graphs the class of -free graphs have bounded clique-width. We then present our main theorem that gives a summary of our current knowledge of those pairs for which the class of -free graphs has bounded clique-width and unbounded clique-width, respectively.111Before finding the combinatorial proof of our main theorem we first obtained a computer-assisted proof using Sage [54] and the Information System on Graph Classes and their Inclusions [23] (which keeps a record of classes for which boundedness or unboundedness of clique-width is known). In particular, we would like to thank Nathann Cohen and Ernst de Ridder for their help. In this way we are able to narrow the gap to 13 open cases (up to some equivalence relation, which we explain later); when we only consider pairs of connected graphs the number of non-equivalent open cases is only two. In order to present our summary, we will need several results from the papers listed above. We will also need these results in Section 6, where we consider graph classes characterized by forbidding a finite family of graphs as subgraphs, minors and topological minors, respectively. For these containment relations we are able to completely determine which of these classes have bounded clique-width.

Algorithmic Consequences

Our results are of interest for any NP-complete problem that is solvable in polynomial time on graph classes of bounded clique-width. In Section 7 we give a concrete application of our results by considering the well-known Colouring problem, which is that of testing whether a graph can be coloured with at most colours for some given integer and which is solvable in polynomial time on any graph class of bounded clique-width [35]. The complexity of Colouring has been studied extensively for -free graphs [19, 21, 28, 36, 42, 52], but a full classification is still far from being settled. Many of the polynomial-time results follow directly from bounding the clique-width in such classes. As such this forms a direct motivation for our research. Another example for which our study might be of interest is the List -Colouring problem (another problem mentioned in the paper of Kobler and Rotics [35]). The complexity of this problem was recently investigated for -free graphs when is a path and is a cycle [33].

Related Work

We finish this section by briefly discussing some related results.

First, a graph class has power-bounded clique-width if there is a constant so that the class consisting of all -th powers of all graphs from has bounded clique-width. Recently, Bonomo, Grippo, Milanič and Safe [2] determined all pairs of connected graphs for which the class of -free graphs has power-bounded clique-width. If a graph class has bounded clique-width, it has power-bounded clique-width. However, the reverse implication does not hold in general. The latter can be seen as follows. Bonomo et al. [2] showed that the class of -free graphs has power-bounded clique-width if and only if is a linear forest (recall that such a class has bounded clique-width if and only if is an induced subgraph of ). Their classification for connected graphs  is the following. Let  be the graph obtained from a 4-vertex star by subdividing one leg times and another leg times. Let be the line graph of . Then the class of -free graphs has power-bounded clique-width if and only if one of the following two cases applies: (i) one of is a path or (ii) one of  is isomorphic to for some and the other one is isomorphic to for some . In particular, the classes of power-unbounded clique-width were already known to have unbounded clique-width.

Second, Kratsch and Schweitzer [37] initiated a study into the computational complexity of the Graph Isomorphism problem (GI) for graph classes defined by two forbidden induced subgraphs. The exact number of open cases is still not known, but Schweitzer [53] very recently proved that this number is finite. There are similarities between classifying the boundedness of clique-width and solving GI for classes of graphs characterized by one or more forbidden induced subgraphs. This was noted by Schweitzer[53], who proved that any graph class that allows a so-called simple path encoding has unbounded clique-width. Indeed, a common technique (see e.g. [34]) for showing that a class of graphs has unbounded clique-width relies on showing that it contains simple path encodings of walls or of graphs in some other specific graph class known to have unbounded clique-width. For -free graphs, GI is polynomial-time solvable if is an induced subgraph of  [14] and GI-complete otherwise [37]. Hence, if only one induced subgraph is forbidden, the dichotomy classifications for clique-width and GI are identical.

2 Preliminaries

Below we define the graph terminology used throughout our paper. For any undefined terminology we refer to Diestel [24].

Let be a graph. The set is the (open) neighbourhood of and is the closed neighbourhood of . The degree of a vertex in a graph is the size of its neighbourhood. The maximum degree of a graph is the maximum vertex degree. For a subset , we let denote the subgraph of induced by , which has vertex set  and edge set . If then, to simplify notation, we may also write instead of . Let  be another graph. We write to indicate that is an induced subgraph of .

Let be a set of graphs. We say that a graph is -free if has no induced subgraph isomorphic to a graph in . If , we may write -free instead of -free. The disjoint union of two vertex-disjoint graphs and is the graph with vertex set and edge set . We denote the disjoint union of copies of by .

For positive integers and , the Ramsey number is the smallest number such that all graphs on vertices contain an independent set of size  or a clique of size . Ramsey’s Theorem [47] states that such a number exists for all positive integers and .

The clique-width of a graph , denoted , is the minimum number of labels needed to construct by using the following four operations:

  1. creating a new graph consisting of a single vertex with label (denoted by );

  2. taking the disjoint union of two labelled graphs and (denoted by );

  3. joining each vertex with label to each vertex with label (, denoted by );

  4. renaming label to (denoted by ).

An algebraic term that represents such a construction of and uses at most  labels is said to be a -expression of (i.e. the clique-width of is the minimum for which has a -expression). For instance, an induced path on four consecutive vertices has clique-width equal to 3, and the following 3-expression can be used to construct it:

Alternatively, any -expression for a graph can be represented by a rooted tree, where the leaves correspond to the operations of vertex creation and the internal nodes correspond to the other three operations. The rooted tree representing the above -expression is depicted in Figure 1. A class of graphs  has bounded clique-width if there is a constant such that the clique-width of every graph in  is at most ; otherwise the clique-width of is unbounded.

\GraphInit\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\SetVertexNormal\Vertex\Edge\Edge\Edge\Edges
Figure 1: The rooted tree representing a 3-expression for .

Let be a graph. The complement of , denoted by , has vertex set and an edge between two distinct vertices if and only if these vertices are not adjacent in .

Let be a graph. We define the following five operations. The contraction of an edge removes  and  from , and replaces them by a new vertex made adjacent to precisely those vertices that were adjacent to or in . By definition, edge contractions create neither self-loops nor multiple edges. The subdivision of an edge replaces by a new vertex  with edges and . Let be a vertex that has exactly two neighbours , and moreover let and be non-adjacent. The vertex dissolution of  removes  and adds the edge . For an induced subgraph , the subgraph complementation operation (acting on with respect to ) replaces every edge present in by a non-edge, and vice versa. Similarly, for two disjoint vertex subsets  and  in , the bipartite complementation operation with respect to and  acts on  by replacing every edge with one end-vertex in  and the other one in by a non-edge and vice versa.

We now state some useful facts for dealing with clique-width. We will use these facts throughout the paper. Let be a constant and let be some graph operation. We say that a graph class is -obtained from a graph class if the following two conditions hold:

  1. every graph in is obtained from a graph in by performing at most times, and

  2. for every there exists at least one graph in obtained from by performing at most  times.

If we do not impose a finite upper bound on the number of applications of then we write that is -obtained from .

We say that preserves boundedness of clique-width if for any finite constant  and any graph class , any graph class that is -obtained from has bounded clique-width if and only if has bounded clique-width.

  1. Vertex deletion preserves boundedness of clique-width [38].

  2. Subgraph complementation preserves boundedness of clique-width [34].

  3. Bipartite complementation preserves boundedness of clique-width [34].

  4. For a class of graphs of bounded maximum degree, let be a class of graphs that is -obtained from , where is the edge subdivision operation. Then  has bounded clique-width if and only if has bounded clique-width [34].

It is easy to show that the condition on the maximum degree in Fact 4 is necessary for the reverse (i.e. the “only if”) direction: for a graph of arbitrarily large clique-width, take a clique  (which has clique-width at most 2) with vertex set , apply an edge subdivision on an edge  in  if and only if is not an edge in and, in order to obtain from this graph, remove any vertex introduced by an edge subdivision (this does not increase the clique-width). As another aside, note that the reverse direction of Fact 4 also holds if we replace “edge subdivisions” by “edge contractions”.222Combine the fact that a class of graphs of bounded maximum degree has bounded clique-width if and only if it has bounded tree-width [31] with the well-known fact that edge contractions do not increase the tree-width of a graph. It was an open problem [30] whether the condition on maximum degree was also necessary in this case. This was recently solved by Courcelle [16], who showed that if  is the class of graphs of clique-width 3 and  is the class of graphs obtained from graphs in  by applying one or more edge contraction operations then  has unbounded clique-width.

We also use a number of other elementary results on the clique-width of graphs. The first one is well known (see e.g. [18]) and straightforward to check.

Lemma 1

The clique-width of a graph with maximum degree at most  is at most .

We also need the well-known notion of a wall. We do not formally define this notion but instead refer to Figure 2, in which three examples of walls of different height are depicted. The class of walls is well known to have unbounded clique-width; see for example [34]. (Note that walls have maximum degree at most 3, hence the degree bound in Lemma 1 is tight.)

\GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Edges\Edges\Edges\Edge\Edge\Edge\Edge\Edge\Edge \GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Edges\Edges\Edges\Edges\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge \GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Edges\Edges\Edges\Edges\Edges\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge\Edge
Figure 2: Walls of height 2, 3, and 4, respectively.

A -subdivided wall is a graph obtained from a wall after subdividing each edge exactly times for some constant .

The following lemma is well known and follows from combining Fact 4 with the aforementioned fact that walls have maximum degree at most 3 and unbounded clique-width.

Lemma 2 ([39])

For any constant , the class of -subdivided walls has unbounded clique-width.

For , the graphs , , denote the cycle, complete graph and path on vertices, respectively, and the graph denotes the star on vertices. The graph  is also called the claw. For , let denote the tree that has only one vertex  of degree and that has exactly three leaves, which are of distance , and from , respectively. Observe that . A graph is said to be a subdivided claw. We let be the class of graphs each connected component of which is either a subdivided claw or a path.

Like Lemma 1, the following lemma is also well known and follows from Lemma 2, by choosing appropriate values for .

Lemma 3 ([39])

Let be a finite set of graphs. If for then the class of -free graphs has unbounded clique-width.

We say that  is bipartite if its vertex set can be partitioned into two (possibly empty) independent sets and . We say that is a bipartition of . Let be a bipartite graph with a fixed partition . A bipartite graph is strongly -free if is -free or else has no bipartition with and such that if and only if for all and . Lozin and Volz [40] characterized all bipartite graphs for which the class of strongly -free bipartite graphs has bounded clique-width. Recently, we proved a similar characterization for -free bipartite graphs; we will use this result in Section 5.

Lemma 4 ([22])

Let be a graph. The class of -free bipartite graphs has bounded clique-width if and only if one of the following cases holds:

  • for some

  • .

From the same paper we will also need the following lemma.

Lemma 5 ([22])

Let . Then is -free if and only if for some integer or is an induced subgraph of one of the graphs in .

We say that a graph is complete multipartite if can be partitioned into independent sets for some integer , such that two vertices are adjacent if and only if they belong to two different sets and . The next result is due to Olariu [44] (the graph is also called the paw).

Lemma 6 ([44])

Every connected -free graph is either complete multipartite or -free.

Every complete multipartite graph has clique-width at most 2. Also, the definition of clique-width directly implies that the clique-width of any graph is equal to the maximum clique-width of its connected components. Hence, Lemma 6 immediately implies the following (well-known) result.

Lemma 7

For any graph , the class of -free graphs has bounded clique-width if and only if the class of -free graphs has bounded clique-width.

Kratsch and Schweitzer [37] proved that the Graph Isomorphism problem is graph-isomorphism complete for the class of -free graphs. It is a straightforward exercise to simplify their construction and use analogous arguments to prove that the class of -free graphs has unbounded clique-width. Recall that Schweitzer [53] proved that any graph class that allows a so-called simple path encoding has unbounded clique-width, implying this result as a direct consequence.

Lemma 8 ([53])

The class of -free graphs has unbounded clique-width.

3 New Classes of Bounded Clique-width

In this section we identify two new graph classes that have bounded clique-width, namely the classes of -free graphs and -free graphs.

We first prove that the class of -free graphs has bounded clique-width. To do so we use a similar approach to that used by Dabrowski, Lozin, Raman and Ries [21] to prove that the classes of -free and -free graphs have bounded clique-width.

Theorem 3.1

The class of -free graphs has bounded clique-width.

Proof

Let be a -free graph. By Lemma 7 we may assume  is -free. Without loss of generality, we may also assume that  is connected (as otherwise we could consider each connected component of  separately). If  is bipartite, then has bounded clique-width by Lemma 4. For the remainder of the proof we assume that is not bipartite, that is, contains an induced odd cycle . Because  is -free, .

First, suppose that . We claim that . Indeed, suppose not. Since  is connected,  must have a vertex that is adjacent to a vertex of . Since is -free, cannot be adjacent to any two consecutive vertices of the cycle . Since is an odd cycle, must therefore have two consecutive non-neighbours on the cycle. Without loss of generality we assume that  is adjacent to and non-adjacent to and . Then must be adjacent to , otherwise would be isomorphic to . Now cannot be adjacent to or , since is -free. However, then would be a , which is a contradiction. Hence, and as such has clique-width at most 4 by Lemma 1.

From now on we assume that . Every vertex not on has at most two neighbours on the cycle, and if it has two, then these neighbours on cannot be consecutive vertices of  (since is -free). We now partition the vertices of not in into sets, depending on their neighbourhood in . We let denote the vertices with no neighbours on the cycle. We let denote the set of all vertices not on the cycle that are adjacent to both and , where subscripts are interpreted modulo 5. We let denote the set of all vertices that are adjacent to  but to no other vertices of . We say that a set or is large if it contains at least two vertices, otherwise we say that it is small. We say that a set in and a set in are consecutive if and are consecutive vertices on , otherwise, we say that they are opposite. Note that each and each is an independent set, since is -free. We now investigate the possible adjacencies between vertices of these sets through a series of eight claims.

  1. is an independent set and every vertex in is adjacent to every vertex in  and . Suppose there is a vertex . Since is connected, there must be a vertex with a neighbour on the cycle. We may assume without loss of generality that is adjacent to , but not to or . Then must be adjacent to , otherwise would be isomorphic to . Hence every vertex in is adjacent to every vertex in and for all . Because of the fact that if is non-empty then some or must also be non-empty and the fact that  is -free, must be an independent set.

  2. If and are opposite then no vertex of  is adjacent to a vertex of . This follows from the fact that any two such vertices have a common neighbour on and the fact that is -free.

  3. If and are consecutive and large then every vertex of is adjacent to every vertex of . Without loss of generality, let . Suppose is not adjacent to . Then is a . Now suppose that is adjacent to , but not to , then is isomorphic to , which is a contradiction.

  4. If and are consecutive then one of them must be empty. Suppose, for contradiction, that there exist vertices and . Then and  are non-adjacent, as is -free. However, then is isomorphic to , which is a contradiction.

  5. If and are opposite and is large then no vertex of has a neighbour in . Let and . If is adjacent to both and , then is isomorphic to . So is adjacent to at most one vertex of , say is adjacent to , but not to . Then is isomorphic to , which is a contradiction.

  6. Every vertex in has at most one non-neighbour in  and vice versa. If has two non-neighbours then the graph is isomorphic to , which is a contradiction. If has two non-neighbours then is isomorphic to , which is again a contradiction.

  7. If and are consecutive and is large then is empty. Without loss of generality, let and . Suppose, for contradiction, that  and . If is adjacent to both  and then is isomorphic to . Without loss of generality, we therefore assume that is not adjacent to . If is not adjacent to then is isomorphic to . If is adjacent to , then is isomorphic to . Hence in all three cases we have a contradiction.

  8. If and are opposite then every vertex of must be adjacent to every vertex of . Without loss of generality, let , , , and . If and are not adjacent, then is isomorphic to , which is not possible.

We now do as follows. First, we remove the vertices of and all small sets  or  if they exist. In this way we remove at most vertices. Hence,  has bounded clique-width if and only if the resulting graph  has bounded clique-width, by Fact 1. We then consider the remaining sets  and  in . We complement the edges between the vertices in and the vertices not in . If  and  are consecutive, we complement the edges between them. If  and  are opposite, we complement the edges between them. Finally, for any pair and , we complement the edges between them. Then has bounded clique-width if and only if the resulting graph has bounded clique-width, by Fact 3. If two vertices are adjacent in , then they must be members of some  and , respectively. By construction, is a (not necessarily perfect) matching. Thus has clique-width at most 2, completing the proof.∎

Next, we prove that the class of -free graphs has bounded clique-width. To do so we first prove Lemma 9, which says that the class of -free graphs has bounded clique-width. We then use this result to prove Theorem 3.2, which says that the larger class of -free graphs also has bounded clique-width. It is also possible to prove Theorem 3.2 by combining very similar arguments to those in the proof of Lemma 9 together with the fact that the class of -free graphs has bounded clique-width (which follows from Theorem 3.1). However, we believe that such a combined proof would be much harder to follow.

Lemma 9

The class of -free graphs has bounded clique-width.

Proof

Let  be a -free graph. By Lemma 7, we may assume  is -free. Let be an arbitrary vertex in . Let and . Since  is -free, must be an independent set. Since  is -free,  must be -free. Then  must have bounded clique-width by Theorem 3.1.

Suppose that . Then we delete and the vertices of and obtain a graph of bounded clique-width, namely . By Fact 1, we find that also has bounded clique-width. Hence we may assume that .

We prove the following claim.

Claim 1. Let with for some . If is complete bipartite, then the clique-width of is bounded by a function of . In particular, this includes the case where is an independent set.

To prove Claim 3, suppose that is complete bipartite. No vertex in has a neighbour in both partition classes of , due to the fact that  is -free. Because is an independent set, this means that is bipartite, in addition to being -free. Hence, has bounded clique-width by Lemma 4. Then by Fact 1, has clique-width bounded by some function of . This proves Claim 3.

We will use Claim 3 later in the proof and now proceed as follows. We fix three arbitrary vertices ; such vertices exist because . Let be three arbitrary vertices of . We will show that at least one of them is adjacent to at least one of , . Because is -free, two of , , are not pairwise adjacent, say . If both and have no neighbour in , then is isomorphic to , a contradiction. Hence, all vertices of except at most two have at least one neighbour in . Then, by Fact 1, we may assume without loss of generality that all vertices of  have at least one neighbour in .

Let consist of those vertices of that are adjacent to . Let consist of those vertices of that are adjacent to but not to . Let . Note that every vertex in is adjacent to but not to or . Moreover, are three independent sets due to the fact that  is -free. If contains at least three vertices, say , then is isomorphic to . Thus . If , then . Moreover, is complete bipartite, because is an independent set. Hence, we may apply Claim 3. From now on we assume that , and similarly, that .

At least one vertex of any pair from must be adjacent to at least one vertex of any triple from ; otherwise these five vertices, together with , induce a subgraph isomorphic to , since and are independent sets and is adjacent to all vertices of and to none of . Fix three vertices . Then at most one vertex of has no neighbours in . Because , this means that at least one of must have at least three neighbours in . By repeating this argument with different choices of , we find that all but at most two vertices in have at least three neighbours in . So, at least six vertices in have at least three neighbours in , and vice versa.

Let be adjacent to at least three vertices of . If is not adjacent to some , then is isomorphic to . Hence, every vertex of  with at least three neighbours in is adjacent to all vertices of . By reversing the roles of and , we find that every vertex in  with at least three neighbours in must be adjacent to all vertices of . Because there are at least six vertices in with at least three neighbours in , and vice versa, we conclude that all vertices of are adjacent to all vertices of , that is, is complete bipartite. Because , we may apply Claim 3 to complete the proof.∎

Theorem 3.2

The class of -free graphs has bounded clique-width.

Proof

Let be a -free graph. By Lemma 7, we may assume  is -free. Suppose that contains a vertex of degree at most 18. If we remove this vertex and its neighbours, we obtain a -free graph, which has bounded clique-width by Lemma 9. Hence,  also has bounded clique-width, by Fact 1. From now on we assume that has minimum degree at least 19 (the reason for choosing this number becomes clear later).

Let . Let and . Note that and fix three arbitrarily-chosen vertices . Let be the set of vertices in that have no neighbour in . We will need the following claim.

Claim 1. .

We prove Claim 3 as follows. Suppose that there are three vertices that are pairwise non-adjacent. Then at least one of must be adjacent to at least one of , as otherwise would be isomorphic to . Hence  is -free. Because  is also -free, we apply Ramsey’s Theorem and find that . This proves Claim 3.

We proceed as follows. Let . Let consist of those vertices of that are adjacent to . Let consist of those vertices of that are adjacent to  but not to . Let . Note that every vertex in is adjacent to , but not to or . Moreover, , , are three independent sets due to the fact that  is -free.

We need the following claim.

Claim 2. Let with , and . Then there exist vertices and such that is a complete bipartite graph minus a matching.

We prove Claim 3 as follows. Suppose and with and . Let and be pairwise distinct. Recall that and are independent sets. Then at least one of must be adjacent to at least one of , as otherwise the graph would be isomorphic to . This means that at most two vertices in have no neighbour in . Hence, as , at least one of has at least three neighbours in . Repeating this argument with different choices of , we find that all but at most two vertices in have at least three neighbours in .

Every vertex that is adjacent to at least three vertices of , say , must be adjacent to all but at most one vertex of , since if is not adjacent to , then would be a . Because all but at most two vertices in have at least three neighbours in , this means that all but at most two vertices of are adjacent to all but at most one vertex of . Because , this means that every vertex of except at most one has at least three neighbours in . Let be this exceptional vertex; if it does not exist then we pick arbitrarily. If , let be three of its neighbours in . Then cannot be non-adjacent to two vertices, say in , otherwise would be a . Thus every vertex in is adjacent to all but at most one vertex of . Since , every vertex in , except at most one has at least three neighbours in and as stated above must therefore be adjacent to all but at most one vertex of . We let denote this exceptional vertex; if it does not exist, then we pick arbitrarily. Because and are independent sets, we conclude that is a complete bipartite graph minus a (not necessarily perfect) matching. If a different pair of sets in both have at least nine vertices, the claim follows by the same arguments.

We now consider three different cases.

Case 1. At least two sets out of have less than nine vertices.
Suppose and . Recall that are independent sets and that is -free. Then is bipartite and -free. Consequently, it has bounded clique-width by Lemma 4. We have by Claim 3. Then . Hence,  has bounded clique-width by Fact 1. If a different pair of sets in both have less than nine vertices, we apply the same arguments.

Case 2. Exactly one set out of has less than nine vertices.
Suppose . Hence and . By Claim 3 we find that there exist two vertices and such that is a complete bipartite graph minus a matching. Let . Suppose, for contradiction, that is adjacent to a vertex and to a vertex . Then is not adjacent to any other vertices of , otherwise would not be -free. Recall that  is an independent set. Hence . We have by Claim 3. Hence, , which is a contradiction since  has minimum degree at least 19. We conclude that no vertex in has neighbours in both and . Because  is independent and is -free, this means that is bipartite and -free. Consequently, it has bounded clique-width by Lemma 4. Because , we conclude that