Bounding the CliqueWidth 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.
Abstract
A graph is free if it has no induced subgraph isomorphic to . We continue a study into the boundedness of cliquewidth of subclasses of perfect graphs. We identify five new classes of free split graphs whose cliquewidth 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 cliquewidth.
1 Introduction
Cliquewidth is a wellstudied 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 cliquewidth if there is a constant such that the cliquewidth of every graph in the class is at most . Much research has been done identifying whether or not various classes have bounded cliquewidth [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 cliquewidth 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 cliquewidth, such as linear cliquewidth [26] and powerbounded cliquewidth [2]. Moreover, the (un)boundedness of the cliquewidth 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 cliquewidth relies on showing that it contains simple path encodings of walls or of graphs in some other specific graph class known to have unbounded cliquewidth. Furthermore, Grohe and Schweitzer [24] recently proved that Graph Isomorphism is polynomialtime solvable on graphs of bounded cliquewidth.
In this paper we continue a study into the boundedness of cliquewidth of subclasses of perfect graphs. Cliquewidth is still a very difficult graph parameter to deal with. For instance, deciding whether or not a graph has cliquewidth at most for some fixed constant is only known to be polynomialtime solvable if [13], but is a longstanding open problem for . Our longterm goal is to increase our understanding of cliquewidth. To this end we aim to identify new classes of bounded cliquewidth. In order to explain some previously known results, along with our new ones, we first give some terminology.
Terminology. For two vertexdisjoint 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).


bull  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 cliquewidth, 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 cliquewidth 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 cliquewidth 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 cliquewidth 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 socalled strongly free bipartite graphs considered in [31]).
Theorem 1.3 ([19])
Let be a graph. The class of free bipartite graphs has bounded cliquewidth 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 cliquewidth [32]. As with the previous graph classes, we forbid one additional induced subgraph . We aim to classify the boundedness of cliquewidth for free split graphs and to identify new graph classes of bounded cliquewidth along the way. Theorem 1.2 also provides motivation, as it would be useful to know whether or not the cliquewidth 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 cliquewidth if and only if the class of free split graphs has bounded cliquewidth (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 cliquewidth if and only if

or is isomorphic to for some ;

or ;

or ;

or ;

or ;

or or

or .




for 






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 socalled 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 cliquewidth then or must be an independent set or an induced subgraph of or . Both of these graphs have seven vertices. The sixvertex induced subgraphs of are: and . The sixvertex 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 cliquewidth 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 selfloops. 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 anticomplete to if every vertex in is nonadjacent to every vertex in . Similarly, a vertex is complete or anticomplete to if it is adjacent or nonadjacent, respectively, to every vertex of . A set of vertices is a module if every vertex not in is either complete or anticomplete to . A module of is trivial if it contains zero, one or all vertices of , otherwise it is nontrivial. 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 CliqueWidth
The cliquewidth of a graph , denoted , is the minimum number of labels needed to construct by using the following four operations:

creating a new graph consisting of a single vertex with label ;

taking the disjoint union of two labelled graphs and ;

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

renaming label to .
A class of graphs has bounded cliquewidth if there is a constant such that the cliquewidth of every graph in is at most ; otherwise the cliquewidth 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 nonedge, 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 endvertex in and the other one in by a nonedge and vice versa.
We now state some useful facts about how the above operations (and some other ones) influence the cliquewidth 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:

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

for every there exists at least one graph in obtained from by performing at most times.
We say that preserves boundedness of cliquewidth if for any finite constant and any graph class , any graph class that is obtained from has bounded cliquewidth if and only if has bounded cliquewidth.
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 cliquewidth if and only if the class of free split graphs has bounded cliquewidth.
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 cliquewidth.
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 blackandwhite 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 blackandwhite 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:

the (unlabelled) graphs and are isomorphic, and

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 blackandwhite 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 nonisomorphic 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.


For a more indepth 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 cliquewidth if and only if one of the following cases holds:

or for some ;

or ;

or

.





for 
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 cliquewidth 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.^{1}^{1}1For 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 cliquewidth, 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 cliquewidth 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 cliquewidth 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 onebyone, 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 cliquewidth.
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 cliquewidth. 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 cliquewidth 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 selfcontained, we include a (much simpler) direct proof for this case.
Theorem 3.3
The class of free split graphs has bounded cliquewidth.
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 nonneighbour 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 nonneighbours of and in (which exist since ). Then one of ’s neighbours in must be or otherwise would be a , a contradiction.
If is is nonempty, 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 nonneighbour 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 cliquewidth at most . This completes the proof.∎
It remains to prove that the class of free graphs has bounded cliquewidth for . We do this in Theorems 3.4–3.6.
Theorem 3.4
The class of free split graphs has bounded cliquewidth.
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 vertexdisjoint 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 cliquewidth by Theorem 3.2.
We may therefore assume that contains vertexdisjoint special bulls, , say. For , let and . In the remainder of the proof, we will show that must contain a nontrivial 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 nonneighbours in then or .
Observation 2. Every has a nonneighbour in every .
Consider the special bulls and . By Observation 3, every vertex in must have a nonneighbour in and a nonneighbour in . Let denote the set of vertices in that are nonadjacent 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 nonadjacent to at least one vertex in each of , so it must have at least nonneighbours in . Let be the set of vertices in that are nonadjacent to and note that since is maximum in , the set is anticomplete 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 anticomplete 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 anticomplete 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 anticomplete to . Therefore is a nontrivial module of , contradicting the fact that is prime. This completes the proof. ∎
Theorem 3.5
The class of free split graphs has bounded cliquewidth.
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 vertexdisjoint 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 cliquewidth.
We may therefore assume that contains two vertexdisjoint 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 nonneighbour . 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 nonneighbour , otherwise would be an . It follows that:
(1) 
Next, by Claim 3, since and have incomparable neighbourhoods and a common nonneighbour in , namely it follows that:
(2) 
Combining (1) and (2), we conclude that:
(3) 
Now and have a common nonneighbour, 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 cliquewidth.
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 vertexdisjoint 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 cliquewidth by Theorem 3.2.
We may therefore assume that contains vertexdisjoint 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 nonneighbour 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 nonneighbour in . Now suppose for contradiction that a vertex has no neighbours in . Let be the other vertex of . It must have a nonneighbour . Note that is then anticomplete 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 nonneighbours 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 nonadjacent 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 cliquewidth 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 cliquewidth. Then must be a split graph, otherwise the class of free split graphs would include all split graphs, in which case the cliquewidth 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 nonadjacent, 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 anticomplete to and complete to . Since and the same is true for . ( and may or may not be adjacent to eachother.) 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 nonadjacent 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 blackandwhite 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 cliquewidth, by Fact 2 it follows that the class of weakly free bipartite graphs must have bounded cliquewidth. By Theorem 2.1, must therefore be a black independent set, a white independent set or a labelled induced subgraph of