Bounding the Clique-Width of {H}-free Split Graphs An extended abstract of this paper appeared in the proceedings of EuroComb 2015 [4]. The research in this paper was supported by EPSRC (EP/K025090/1). The third author is grateful for the generous support of the Graduate (International) Research Travel Award from Simon Fraser University and Dr. Pavol Hell’s NSERC Discovery Grant.

Bounding the Clique-Width of -free Split Graphs thanks: An extended abstract of this paper appeared in the proceedings of EuroComb 2015 [4]. The research in this paper was supported by EPSRC (EP/K025090/1). The third author is grateful for the generous support of the Graduate (International) Research Travel Award from Simon Fraser University and Dr. Pavol Hell’s NSERC Discovery Grant.

Andreas Brandstädt Institute of Computer Science, Universität Rostock,
Albert-Einstein-Straße 22, 18059 Rostock, Germany
ab@informatik.uni-rostock.de
   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
  
Shenwei Huang
School of Computing Science, Simon Fraser University,
8888 University Drive, Burnaby B.C., V5A 1S6, Canada
shenweih@sfu.ca
   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

A graph is -free if it has no induced subgraph isomorphic to . We continue a study into the boundedness of clique-width of subclasses of perfect graphs. We identify five new classes of -free split graphs whose clique-width is bounded. 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  for which the class of -free split graphs has bounded clique-width.

1 Introduction

Clique-width is a well-studied graph parameter; see for example the surveys of Gurski [25] and Kamiński, Lozin and Milanič [27]. A graph class is said to be of bounded clique-width if there is a constant  such that the clique-width of every graph in the class is at most . 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, 19, 18, 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. In a recent series of papers [3, 16, 18] the clique-width of graph classes characterized by two forbidden induced subgraphs was investigated. In particular we refer to [18] for details on how new results can be combined with known results to give a classification for all but  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]. Moreover, the (un)boundedness of the clique-width of a graph class seems to be related to the computational complexity of the Graph Isomorphism problem, which has in particular been investigated for graph classes defined by two forbidden induced subgraphs [28, 33]. Indeed, a common technique (see e.g. [27]) 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. Furthermore, Grohe and Schweitzer [24] recently proved that Graph Isomorphism is polynomial-time solvable on graphs of bounded clique-width.

In this paper we continue a study into the boundedness of clique-width of subclasses of perfect graphs. Clique-width is still a very difficult graph parameter to deal with. For instance, deciding whether or not a graph has clique-width at most  for some fixed constant  is only known to be polynomial-time solvable if  [13], but is a long-standing open problem for . Our long-term goal is to increase our understanding of clique-width. To this end we aim to identify new classes of bounded clique-width. In order to explain some previously known results, along with our new ones, we first give some terminology.

Terminology. For two vertex-disjoint graphs  and , the disjoint union is denoted by  and the disjoint union of  copies of  is denoted by . The complement of a graph , denoted by , has vertex set and an edge between two distinct vertices if and only if these vertices are not adjacent in . For two graphs  and  we write to indicate that  is an induced subgraph of . The graphs and  denote the cycle, complete graph, star and path on  vertices, respectively. The graph , for , denotes the subdivided claw, that is the tree that has only one vertex  of degree  and exactly three leaves, which are of distance  and  from , respectively. For a set of graphs , a graph  is -free if it has no induced subgraph isomorphic to a graph in . The bull is the graph with vertices and edges ; the dart is the graph obtained from the bull by adding the edge  (see also Figure 1).

\GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex\Edges\Edges \GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex\Edges
bull dart
Figure 1: The bull and the dart.

A graph  is perfect if, for every induced subgraph , the chromatic number of  equals its clique number. By the Strong Perfect Graph Theorem [12], a graph  is perfect if and only if both  and  are -free. A graph  is chordal if it is -free and weakly chordal if both  and  are -free. Every split graph is chordal, every chordal graph is weakly chordal and every weakly chordal graph is perfect.

Known Results on Subclasses of Perfect Graphs. We start off with the following known theorem, which shows that the restriction of -free graphs to -free weakly chordal graphs does not yield any new graph classes of bounded clique-width, as both classifications are exactly the same.

Theorem 1.1 ([3, 18])

Let  be a graph. The class of -free (weakly chordal) graphs has bounded clique-width if and only if  is an induced subgraph of .

Motivated by Theorem 1.1 we investigated classes of -free chordal graphs in an attempt to identify new classes of bounded clique-width and as a (successful) means to find reductions to solve more cases in our classification for -free graphs. This classification for classes of -free chordal graphs is almost complete except for two cases, which we call  and  (see Figure 2 for a definition).

Theorem 1.2 ([3])

Let  be a graph not in . The class of -free chordal graphs has bounded clique-width if and only if

  • for some ;

  • ;

  • ;

  • ;

  • ;

  • ;

  • or

  • .

In contrast to chordal graphs, the classification for bipartite graphs, another class of perfect graphs, is complete. This classification was used in the proof of Theorem 1.2 and it is similar to a characterization of Lozin and Volz [31] for a different variant of the notion of -freeness in bipartite graphs (see [19] for an explanation of the difference between -free bipartite graphs and the so-called strongly -free bipartite graphs considered in [31]).

Theorem 1.3 ([19])

Let  be a graph. The class of -free bipartite graphs has bounded clique-width if and only if

  • for some ;

  • ;

  • ;

  • or

  • .

Our Results. We consider subclasses of split graphs. A graph is a split graph if it has a split partition, that is, a partition of  into two (possibly empty) sets  and , where  is a clique and  is an independent set. The class of split graphs coincides with the class of -free graphs [22] and is known to have unbounded clique-width [32]. As with the previous graph classes, we forbid one additional induced subgraph . We aim to classify the boundedness of clique-width for -free split graphs and to identify new graph classes of bounded clique-width along the way. Theorem 1.2 also provides motivation, as it would be useful to know whether or not the clique-width of -free split graphs is bounded when or (the two missing cases for chordal graphs; recall that chordal graphs form a superclass of split graphs). We give affirmative answers for both of these cases. It should be noted that, for any graph  the class of -free split graphs has bounded clique-width if and only if the class of -free split graphs has bounded clique-width (see also Lemma 1). As such our main result shows that there are only two open cases (see also Figs. 2 and 3).

Theorem 1.4

Let  be a graph such that neither  nor  is in . The class of -free split graphs has bounded clique-width if and only if

  • or  is isomorphic to  for some ;

  • or ;

  • or ;

  • or ;

  • or ;

  • or or

  • or .

\GraphInit\SetVertexSimple\Vertices \GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Edges\Edges \GraphInit\SetVertexSimple\Vertices\Vertex\Vertex\Edges\Edges\Edges\Edges
 for 
\GraphInit\SetVertexSimple\Vertices\Vertex\Vertex\Edges\Edges\Edges\Edges \GraphInit\SetVertexSimple\Vertices\Vertex\Vertex\Edges\Edges\Edges \GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Edges\Edge \GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Edges\Edges
Figure 2: The graphs  from Theorem 1.4 for which the classes of -free split graphs and -free split graphs have bounded clique-width.
\GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Vertex\Edges \GraphInit\SetVertexSimple\Vertices\Vertex\Vertex\Vertex\Edges\Edges\Edges\Edges
Figure 3: The (only) two graphs for which it is not known whether or not the classes of -free split graphs and -free split graphs have bounded clique-width.

In Section 3 we prove each of the bounded cases in Theorem 1.4. These proofs use results from the literature, which we state in Section 2, together with some other preliminaries. In particular, we will exploit the close relationship between -free split graphs and so-called weakly -free bipartite graphs (see the next section for a definition). This enables us to apply Theorem 2.1 (a variant of Theorem 1.3; both these theorems were proved in [19]) after first transforming a split graph into a bipartite graph by removing the edges of the clique (this has to be done carefully, as a graph may have multiple split partitions).

In Section 4 we prove Theorem 1.4. We show that if the class of -free split graphs has bounded clique-width then  or  must be an independent set or an induced subgraph of  or . Both of these graphs have seven vertices. The six-vertex induced subgraphs of  are: and . The six-vertex induced subgraphs of  are: and . These graphs and their complements are precisely the cases listed in Theorem 1.4 (and for which we prove boundedness in Section 3). Hence, we can also formulate our main theorem as follows.

Theorem 1.4 (alternative formulation). Let  be a graph such that neither  nor  is in . The class of -free split graphs has bounded clique-width if and only if

  • or  is isomorphic to  for some ;

  • or or

  • or .

2 Preliminaries

We only consider graphs that are finite, undirected and have neither multiple edges nor self-loops. In this section we define some more graph terminology, additional notation and give some known lemmas from the literature that we will need to prove our results. We refer to the textbook of Diestel [21] for any undefined terminology.

Let be a graph. The set is the neighbourhood of . The degree of a vertex in  is the size  of its neighbourhood. Let  with . Then  is complete to  if every vertex in  is adjacent to every vertex in , and  is anti-complete to  if every vertex in  is non-adjacent to every vertex in . Similarly, a vertex is complete or anti-complete to  if it is adjacent or non-adjacent, respectively, to every vertex of . A set  of vertices is a module if every vertex not in  is either complete or anti-complete to . A module of  is trivial if it contains zero, one or all vertices of , otherwise it is non-trivial. A graph  is prime if every module in  is trivial. We say that a vertex  distinguishes two vertices  and  if  is adjacent to precisely one of  and . Note that if a set  is not a module then there must be vertices and a vertex such that  distinguishes  and .

In a partially ordered set , two elements are comparable if or , otherwise they are incomparable. A set is a chain if the elements of  are pairwise comparable.

2.1 Clique-Width

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 ;

  2. taking the disjoint union of two labelled graphs  and ;

  3. joining each vertex with label  to each vertex with label  ();

  4. renaming label  to .

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.

Let  be a graph. We define the following operations. 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 about how the above operations (and some other ones) influence the clique-width of a graph. 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.

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 [29].

  2. Subgraph complementation preserves boundedness of clique-width [27].

  3. Bipartite complementation preserves boundedness of clique-width [27].

Combining the fact that the complement of any split graph is split with Fact 2 leads to the following lemma.

Lemma 1

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

We will also need the following two results.

Lemma 2 ([14])

If  is the set of all prime induced subgraphs of a graph  then .

Lemma 3 ([32])

The class of split graphs has unbounded clique-width.

2.2 Bipartite Graphs

A graph is bipartite if its vertex set can be partitioned into two (possibly empty) independent sets. Let  be a bipartite graph. A black-and-white labelling  of  is a labelling that assigns either the colour “black” or the colour “white” to each vertex of  in such a way that the two resulting monochromatic colour classes  and  form a bipartition of  into two (possibly empty) independent sets. We say that  is a labelled bipartite graph if we are also given a fixed black-and-white labelling. We denote a graph  with such a labelling  by . It is important to note that the pair is ordered, that is, and are different labelled bipartite graphs. Two labelled bipartite graphs  and  are isomorphic if the following two conditions hold:

  1. the (unlabelled) graphs  and  are isomorphic, and

  2. there exists an isomorphism such that for all , it holds that if and only if .

Moreover, in this case  and  are said to be isomorphic labellings. We write if , and . In this case we say that  is a labelled induced subgraph of . Note that the two labelled bipartite graphs  and  are isomorphic if and only if  is a labelled induced subgraph of , and vice versa.

If  is a bipartite graph with a labelling , we let  denote the “opposite” labelling labelling to , namely the labelling obtained from  by reversing the colours. If  is a bipartite graph with the property that among all its black-and-white labellings, all those that maximize the number of black vertices are isomorphic, then we pick one such labelling and call it . If such a unique labelling  does exist, we let  denote the opposite labelling to .

Let  be an (unlabelled) bipartite graph, and let  be a labelled bipartite graph. Then  is weakly -free if there is a labelling  of  such that  does not contain  as a labelled induced subgraph. Similarly, let be a set of labelled bipartite graphs. Then  is weakly -free if there is a labelling  of  such that  does not contain any graph in as a labelled induced subgraph.

Example. The two non-isomorphic labelled bipartite graphs corresponding to  are shown in Figure 4. Every edgeless graph is weakly -free and weakly -free (simply label all the vertices white or all the vertices black, respectively). However, if a bipartite graph is weakly -free then it cannot contain any vertices. Hence, a bipartite graph can be weakly -free,, weakly -free, while not being weakly -free.

\GraphInit\SetVertexSimple\Vertex \GraphInit\SetVertexSimple\Vertex
Figure 4: The two pairwise non-isomorphic labellings of .

For a more in-depth discussion of weakly -free bipartite graphs we refer to [19]. In this paper we will make use of the following theorem (see also Figure 5).

Theorem 2.1 ([19])

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

  • or for some ;

  • or ;

  • or

  • .

\GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\Vertex \GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\Vertex\SetVertexSimple\Vertex\Vertex\Edges \GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\SetVertexSimple\Vertex\Vertex\Vertex\Edge\Edges \GraphInit\SetVertexSimple\Vertex\Vertex\Vertex\SetVertexSimple\Vertex\Vertex\Vertex\Edges
for
Figure 5: The labelled bipartite graphs from Theorem 2.1.

Similarly to the way that a bipartite graph can have multiple labellings, a split graph  may have multiple split partitions, say and . We say that two such split partitions are isomorphic if there is an isomorphism of  such that if and only if . Let  and  be split graphs with split partitions and , respectively. Then contains if , and . We will explore the properties of split partitions in the proof of Lemma 4.

3 Proofs of the Bounded Cases in Theorem 1.4

In this section we show that the clique-width of each of the seven classes of -free graphs given in Theorem 1.4 is bounded. We start with the case , for which we give an explicit bound.111For the other bounded cases we do not specify any upper bounds. This would complicate our proofs for negligible gain, as our primary goal is to show boundedness. Moreover, in our proofs we apply graph operations that may exponentially increase the upper bound on the clique-width, which means that any bounds obtained from our proofs would be very large and far from being tight. Furthermore, we make use of other results that do not give explicit bounds.

Theorem 3.1

For any , the class of -free graphs has clique-width at most .

Proof

Let for some and let  be an -free split graph with split partition . It follows that . In this case it is easy to see that the clique-width of  is at most : We introduce the (at most ) vertices of  with distinct labels. We use one more label for “new” vertices of  and one more label for “processed” vertices of . We then add each vertex of  one-by-one, labelling it with the “new” label, and immediately connect it to all the already “processed” vertices of , along with any relevant vertices of , after which we relabel the new vertex to be “processed.”∎

We now consider the cases and . In order to prove these two cases we apply Theorem 2.1 for the first time.

Theorem 3.2

The class of -free split graphs and the class of -free split graphs have bounded clique-width.

Proof

Let  be or  and let  be the labelled bipartite graph or , respectively. Suppose  is an -free split graph and fix a split partition of . Let  be the graph obtained from  by applying a complementation to . By Fact 2, we need only show that  has bounded clique-width. Now  is a bipartite graph with bipartition . If we label the vertices of  white and the vertices of  black, then we find that  is a weakly -free bipartite graph and therefore has bounded clique-width by Theorem 2.1. ∎

The next theorem follows from Theorem 1.2 and Lemma 1 (recall that every split graph is chordal). However, the proof of the corresponding case for chordal graphs is much more complicated. In light of this, and to make this paper more self-contained, we include a (much simpler) direct proof for this case.

Theorem 3.3

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

Proof

Let  be a -free split graph and fix of a partition of its vertices into a clique  and an independent set . If then  is -free (at most one vertex of any independent set in  can belong to ), in which case we are done by Theorem 3.1. We therefore assume that . Since  is -free, every vertex in  has either at most two neighbours in  or at most one non-neighbour in . Let  be the set of vertices in  that have exactly two neighbours in . Suppose and let  and  be the two neighbours of  in  and let  and  be two common non-neighbours of  and  in  (which exist since ). Then one of ’s neighbours in  must be  or  otherwise would be a , a contradiction.

If  is is non-empty, choose arbitrarily and delete both neighbours of  in  (we may do this by Fact 1) to obtain a graph . Now every vertex of  has at most one neighbour in in the graph . (If  was already empty, then we set .) In the graph  every vertex in  has either at most one neighbour or at most one non-neighbour in . Let  be the set of vertices that have more than one neighbour in . By Fact 3, we may apply a bipartite complementation between  and  to obtain a graph  in which every vertex of  has at most one neighbour in . Finally apply a complementation to the set  (we may do this by Fact 2). The resulting graph is a disjoint union of stars, so it has clique-width at most . This completes the proof.∎

It remains to prove that the class of -free graphs has bounded clique-width for . We do this in Theorems 3.43.6.

Theorem 3.4

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

Proof

Let  be an -free split graph. Fix a split partition of . By Lemma 2, we may assume that  is prime. If  contains an induced bull (see also Figure 1) that has three vertices in  and two in , we say that this bull is special.

First suppose that  does not contain  vertex-disjoint special bulls. By Fact 1, we may delete at most vertices from  to obtain a split graph with no special bulls. Since the resulting graph contains no special bulls, it must be -free, and therefore has bounded clique-width by Theorem 3.2.

We may therefore assume that  contains  vertex-disjoint special bulls, , say. For , let and . In the remainder of the proof, we will show that  must contain a non-trivial module, contradicting the fact that  is prime.

We first state the following two observations, both of which follow directly from the fact that  is an -free split graph.

Observation 1. If have two common non-neighbours in  then or .

Observation 2. Every has a non-neighbour in every .

Consider the special bulls  and . By Observation 3, every vertex in  must have a non-neighbour in  and a non-neighbour in . Let  denote the set of vertices in  that are non-adjacent to both  and , for . (Note that every vertex of  must be in at least one set , but it may be in more than one such set.) By Observation 3, for any two vertices in any set  either or .

Since  is prime, no two vertices of  have the same neighbourhood. We may therefore define a partial order  on : given two vertices , we say that if . Note every set  is a chain under this partial order, so  can be covered by at most nine chains.

We rename the sets  to be , in an arbitrary order, deleting any sets  that are empty, so . For , let  be the maximum element of  (under the  ordering). From the definition of the sets it follows that for , . If there are distinct such that then , so we may delete the set  from the set of chains  that we consider and every vertex of  will still be in some set . In other words, we may assume that are chains under the  ordering, with maximal elements , respectively, where and every pair is incomparable under the  ordering. (Note that , since have incomparable neighbourhoods.)

By Observation 3, for each , the vertex  must be non-adjacent to at least one vertex in each of , so it must have at least  non-neighbours in . Let  be the set of vertices in  that are non-adjacent to  and note that since  is maximum in , the set  is anti-complete to .

Since for the vertices  and  are incomparable, it follows that  is adjacent to all but at most one vertex of  (by Observation 3). Therefore, there is a subset with such that  is complete to , so for .

Claim 1. For there is a vertex such that every vertex in  is either complete or anti-complete to .

We prove Claim 3 as follows. Let  be the smallest (with respect to ) vertex in  that has a neighbour, say , in . Any vertex with is anti-complete to . Now , since  is not a neighbour of  and , since and . By Observation 3, contains all but at most one vertex of . If there is a vertex then  is complete to (if there is no such vertex then we choose arbitrarily, and the same conclusion holds). If and then , as desired. This completes the proof of Claim 3.

Recall that for , . Let (where are defined as in Claim 3 above). Then and every vertex in  is either complete or anti-complete to . Therefore  is a non-trivial module of , contradicting the fact that  is prime. This completes the proof. ∎

Theorem 3.5

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

Proof

Let  be an -free split graph. Fix a split partition of . By Lemma 2, we may assume that  is prime. If  contains an induced  (see also Figure 2) it must have three vertices in  and three in  (since  has a unique split partition).

First suppose that  does not contain two vertex-disjoint copies of . By Fact 1, we may delete at most six vertices from  to obtain a -free split graph. By Theorem 3.2, the resulting graph (and thus ) has bounded clique-width.

We may therefore assume that  contains two vertex-disjoint copies of , say  and . For , let and , where .

We say that two vertices have comparable neighbourhoods if or . Otherwise we say that  and  have incomparable neighbourhoods.

Claim 1. Suppose have a common non-neighbour . If  and  have incomparable neighbourhoods then .

We proof Claim 3 as follows. Since  and  have incomparable neighbourhoods, there must be a vertex and a vertex . Suppose, for contradiction, that there is another vertex . Then is an . This contradiction completes the proof of Claim 3.

The vertices  and  cannot have a common non-neighbour , otherwise would be an . It follows that:

(1)

Next, by Claim 3, since  and  have incomparable neighbourhoods and a common non-neighbour in , namely  it follows that:

(2)

Combining (1) and (2), we conclude that:

(3)

Now  and  have a common non-neighbour, namely . Note that . By Claim 3 it follows that either (if  and  have comparable neighbourhoods) or (if they do not). This means that is a subset of or . In particular, or , respectively, is complete to . Then this vertex, together with , and  induces an  in . This contradiction completes the proof.∎

Theorem 3.6

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

Proof

Let  be an -free split graph. Fix a split partition of . By Lemma 2, we may assume that  is prime. If  contains an induced dart (see also Figure 1) which has has three vertices in  and two in , we say that this dart is special.

First suppose that  does not contain  vertex-disjoint special darts. By Fact 1, we may delete at most vertices from  to obtain a split graph with no special dart. Since the resulting graph contains no special copies of the dart, it must be -free, and therefore has bounded clique-width by Theorem 3.2.

We may therefore assume that  contains  vertex-disjoint special darts, , say. For , let and . We will use the following claim.

Claim 1. If then every vertex of  has at least one neighbour and at least one non-neighbour in .

We prove Claim 3 as follows. If then the claim follows from the definition of . Suppose . If a vertex is complete to  then is an , which is a contradiction. Therefore each vertex in  has at least one non-neighbour in . Now suppose for contradiction that a vertex has no neighbours in . Let  be the other vertex of . It must have a non-neighbour . Note that  is then anti-complete to . Now is an . This contradiction completes the proof of Claim 3.

Claim 3 implies that for every , every vertex of  must have one of the six possible neighbourhoods in , namely those that contain at least one vertex of , but not all vertices of . This means we can partition the vertices of into 36 sets (some of which may be empty), according to their neighbourhood in . Since consists of 38 vertices, two of these vertices, say  and  must have the same neighbourhood in . Furthermore, by Claim 3, they have a common neighbour  and common non-neighbours and . Since the graph  is prime, the set cannot be a module. Therefore there must be a vertex  that distinguishes  and , say  is adjacent to , but non-adjacent to . Note that , so it must be adjacent to and . Now is an . This contradiction completes the proof.∎

4 Completing the Proof of Theorem 1.4

In this section we use the results from the previous section to prove our main result. We also need the following lemma.

Lemma 4 (Key Lemma)

If the class of -free split graphs has bounded clique-width then  or  is isomorphic to  for some  or is an induced subgraph of  or .

Proof

Suppose that  is a graph such that the class of -free split graphs has bounded clique-width. Then  must be a split graph, otherwise the class of -free split graphs would include all split graphs, in which case the clique-width would be unbounded by Lemma 3.

Suppose that  has two split partitions and that are not isomorphic. There cannot be two distinct vertices , as then , so they would have to be non-adjacent, and similarly , so they would have to be adjacent, a contradiction. Hence, . For the same reason, .

Next suppose that . Then there exist vertices and . Let and . Then and . Since  and , must be anti-complete to  and complete to . Since and the same is true for . ( and  may or may not be adjacent to each-other.) However, this means that and are isomorphic split partitions of , which is a contradiction.

Due to the above, we may assume without loss of generality that and . Hence there is a vertex  such that and . Let and note that  has split partition (though  may also have a different split partition) and that  can be obtained from  by adding a vertex that is adjacent to every vertex of  and non-adjacent to every vertex of .

Let  be the labelled bipartite graph obtained from  by complementing , colouring every vertex of  white and every vertex of  black. Let  be a weakly -free graph. Then has a black-and-white labelling such that does not contain  as a labelled induced subgraph. Let  be the split graph obtained from  by complementing the set of black vertices. Then is a split partition of  that does not contain . Therefore, has a split partition that does not contain or . Hence, is -free. Since we assumed that the class of -free split graphs has bounded clique-width, by Fact 2 it follows that the class of weakly -free bipartite graphs must have bounded clique-width. By Theorem 2.1, must therefore be a black independent set, a white independent set or a labelled induced subgraph of