A graph class is well-quasi-ordered by some containment relation if for any infinite sequence of graphs in , there is a pair with such that is contained in . A graph class has bounded clique-width if there exists a constant such that every graph in has clique-width at most . Both being well-quasi-ordered and having bounded clique-width are highly desirable properties of graph classes in the areas of discrete mathematics and theoretical computer science. To illustrate this, let us mention the seminal project of Robertson and Seymour on graph minors that culminated in 2004 in the proof of Wagner’s conjecture, which states that the set of all finite graphs is well-quasi-ordered by the minor relation. As an algorithmic consequence, given a minor-closed graph class, it is possible to test in cubic time whether a given graph belongs to this class. The algorithmic importance of having bounded clique-width follows from the fact that many well-known NP-hard problems, such as Graph Colouring and Hamilton Cycle, become polynomial-time solvable for graph classes of bounded clique-width (this follows from combining results from several papers [6, 16, 19, 27] with a result of Oum and Seymour ).
Courcelle  proved that the class of graphs obtained from graphs of clique-width via one or more edge contractions has unbounded clique-width. Hence the clique-width of a graph can be much smaller than the clique-width of its minors. On the other hand, the clique-width of a graph is at least the clique-width of any of its induced subgraphs (see, for example, ). We therefore focus on hereditary classes, that is, on graph classes that are closed under taking induced subgraphs. It is readily seen that a class of graphs is hereditary if and only if it can be characterised by a unique set of minimal forbidden induced subgraphs. Our underlying research goal is to increase our understanding of the relation between well-quasi-orders and clique-width of hereditary graph classes.
As a start, we note that the hereditary class of graphs of degree at most is not well-quasi-ordered by the induced subgraph relation, as it contains the class of cycles, which form an infinite antichain. As every graph of degree at most has clique-width at most , having bounded clique-width does not imply being well-quasi-ordered by the induced subgraph relation. In 2010, Daligault, Rao and Thomassé  asked about the reverse implication: does every hereditary graph class that is well-quasi-ordered by the induced subgraph relation have bounded clique-width? In 2015, Lozin, Razgon and Zamaraev  gave a negative answer. As the set of minimal forbidden induced subgraphs in their counter-example is infinite, the question of Daligault, Rao and Thomassé  remains open for finitely defined hereditary graph classes, that is, hereditary graph classes for which is finite.
Conjecture 1 ()
If a finitely defined hereditary class of graphs is well-quasi-ordered by the induced subgraph relation, then has bounded clique-width.
If Conjecture 1 is true, then for finitely defined hereditary graph classes the aforementioned algorithmic consequences of having bounded clique-width also hold for the property of being well-quasi-ordered by the induced subgraph relation. A hereditary graph class defined by a single forbidden induced subgraph is well-quasi-ordered by the induced subgraph relation if and only if it has bounded clique-width if and only if is an induced subgraph of (see, for instance, [12, 14, 21]). Hence Conjecture 1 holds when has size . We consider the case when has size , say . Such graph classes are said to be bigenic or -free graph classes. In this case Conjecture 1 is also known to be true except for two stubborn open cases, namely and ; see .
Our Results. In Section 4, we prove that Conjecture 1 holds for the class of -free graphs by showing that the class of -free graphs has bounded clique-width and is well-quasi-ordered by the induced subgraph relation. We do this by using a general technique explained in Section 3. This technique is based on extending (a labelled version of) well-quasi-orderability or boundedness of clique-width of the bipartite graphs in a hereditary graph class to a special subclass of -partite graphs in . The crucial property of these -partite graphs is that no three vertices from the three different partition classes form a clique or independent set. We say that such -partite graphs curious. A more restricted version of this concept was used to prove that -free graphs have bounded clique-width . In Section 4 we show how to generalise results for curious -free graphs to the whole class of -free graphs. In the same section we also show how to apply our technique to prove that the class of -free graphs is well-quasi-ordered (it was already known  that the class of -free graphs has bounded clique-width). We note that our results also imply that the class of -free graphs is well-quasi-ordered, which was not previously known (see [2, 21]). See also Figure 1 for pictures of the forbidden induced subgraphs mentioned in this paragraph.
Dichotomies. Previously, well-quasi-orderability was known for -free graphs , -free bipartite graphs  and -free bipartite graphs . It has also been shown that -free bipartite graphs are not well-quasi-ordered if contains an induced ,  or . Moreover, for every , the class of -free graphs is finite due to Ramsey’s Theorem . The above results lead to the following known dichotomy for -free bipartite graphs.
Let be a graph. The class of -free bipartite graphs is well-quasi-ordered by the induced subgraph relation if and only if for some or is an induced subgraph of , or .
Now combining the aforementioned known results for -free graphs and -free bipartite graphs with our new results yields the following new dichotomy for -free triangle-free graphs, which is exactly the same as the one in Theorem 1.1.
Let be a graph. The class of -free graphs is well-quasi-ordered by the induced subgraph relation if and only if for some or is an induced subgraph of , , or .
Besides our technique based on curious graphs, we also expect that Theorem 1.2 will itself be a useful ingredient for showing results for other graph classes, just as Theorem 1.1 has already proven to be useful (see e.g. ).
For clique-width the following dichotomy is known for -free bipartite graphs.
Theorem 1.3 ()
Let be a graph. The class of -free bipartite graphs has bounded clique-width if and only if for some or is an induced subgraph of , , or .
It would be interesting to determine whether -free graphs allow the same dichotomy with respect to the boundedness of their clique-width. The evidence so far is affirmative, but in order to answer this question two remaining cases need to be solved, namely and ; see Section 2 for the definition of the graph . Both cases turn out to be highly non-trivial; in particular, the class of -free graphs contains the class of -free graphs, and the class of -free graphs contains both the classes of -free and -free graphs.
In Section 5 we give state-of-the-art summaries for well-quasi-orderability and boundedness of clique-width of bigenic graph classes (which include our new results), together with an overview of the missing cases for both problems (which include the missing cases mentioned in this section).
Throughout the paper, we consider only finite, undirected graphs without multiple edges or self-loops. Below, we define further graph terminology.
The disjoint union of two vertex-disjoint graphs and is denoted by and the disjoint union of copies of a graph is denoted by . The complement of a graph has vertex set and an edge between two distinct vertices if and only if . 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 . We use to denote the graph obtained from by deleting every vertex in , that is, . 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 graphs and are also called the triangle and claw, respectively. A graph is a linear forest if every component of is a path (on at least one vertex). 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. Observe that . We let denote the class of graphs, each connected component of which is either a subdivided claw or a path.
For a set of graphs , a graph is -free if it has no induced subgraph isomorphic to a graph in ; if , we may write -free instead of -free. A graph is -partite if its vertex can be partitioned into (possibly empty) independent sets; -partite graphs are also known as bipartite graphs. The biclique or complete bipartite graph is the bipartite graph with sets in the partition of size and respectively, such that every vertex in one set is adjacent to every vertex in the other set. For a graph , the set denotes the neighbourhood of . Let be a set of vertices in . A vertex is complete to if it is adjacent to every vertex of and anti-complete to if it is non-adjacent to every vertex of . A set of vertices is complete (resp. anti-complete) to if every vertex in is complete (resp. anti-complete) to . If and are disjoint sets of vertices in a graph, we say that the edges between these two sets form a matching if each vertex in has at most one neighbour in and vice versa (if each vertex has exactly one such neighbour, we say that the matching is perfect). Similarly, the edges between these sets form a co-matching if each vertex in has at most one non-neighbour in and vice versa. We say that the set dominates if every vertex of has at least one neighbour in . Similarly, a vertex dominates if every vertex of is adjacent to . A vertex distinguishes if has both a neighbour and a non-neighbour in . The set is a module of if no vertex in distinguishes . A module is non-trivial if , otherwise it is trivial. A graph is prime if it has only trivial modules. Two vertices are false twins if they have the same neighbourhood (note that such vertices must be non-adjacent). Clearly any prime graph on at least three vertices cannot contain a pair of false twins, as any such pair of vertices would form a non-trivial module.
We will use the following structural result.
Lemma 1 ()
Let be a connected -free graph that does not contain a pair of false twins. Then is either bipartite or a cycle.
The clique-width of a graph 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 clique-width if there is a constant such that the clique-width of every graph in is at most ; otherwise the clique-width 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:
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 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.
Lemma 2 ()
Let be a graph and let be the set of all induced subgraphs of that are prime. Then .
A quasi order on a set is a reflexive, transitive binary relation. Two elements in this quasi-order are comparable if or , otherwise they are incomparable. A set of elements in a quasi-order is a chain if every pair of elements is comparable and it is an antichain if every pair of elements is incomparable. The quasi-order is a well-quasi-order if any infinite sequence of elements in contains a pair with and . Equivalently, a quasi-order is a well-quasi-order if and only if it has no infinite strictly decreasing sequence and no infinite antichain.
For an arbitrary set , let denote the set of finite sequences of elements of . A quasi-order on defines a quasi-order on as follows: if and only if there is a sequence of integers with such that for . We call the subsequence relation.
Lemma 3 (Higman’s Lemma )
If is a well-quasi-order then is a well-quasi-order.
To define the notion of labelled induced subgraphs, let us consider an arbitrary quasi-order . We say that is a labelled graph if each vertex of is equipped with an element (the label of ). Given two labelled graphs and , we say that is a labelled induced subgraph of if is isomorphic to an induced subgraph of and there is an isomorphism that maps each vertex of to a vertex of with . Clearly, if is a well-quasi-order, then a graph class cannot contain an infinite sequence of labelled graphs that is strictly-decreasing with respect to the labelled induced subgraph relation. We therefore say that a graph class is well-quasi-ordered by the labelled induced subgraph relation if it contains no infinite antichains of labelled graphs whenever is a well-quasi-order. Such a class is readily seen to also be well-quasi-ordered by the induced subgraph relation.
Daligault, Rao and Thomassé  showed that every hereditary class of graphs that is well-quasi-ordered by the labelled induced subgraph relation is defined by a finite set of forbidden induced subgraphs. Korpelainen, Lozin and Razgon  conjectured that if a hereditary class of graphs is defined by a finite set of forbidden induced subgraphs, then is well-quasi-ordered by the induced subgraph relation if and only if it is well-quasi-ordered by the labelled induced subgraph relation. Brignall, Engen and Vatter  recently found a class with 14 forbidden induced subgraphs that is a counterexample for this conjecture, that is is well-quasi-ordered by the induced subgraph relation but not by the labelled induced subgraph relation. However, so far, all known results for bigenic graph classes, including those in this paper, verify the conjecture for bigenic graph classes.
Similarly to the notion of preserving boundedness of clique-width, we say that a graph operation preserves well-quasi-orderability by the labelled induced subgraph relation if for any finite constant and any graph class , any graph class that is -obtained from is well-quasi-ordered by this relation if and only if is.
 The following operations preserve well-quasi-orderability by the labelled induced subgraph relation:
Bipartite complementation and
Lemma 5 ()
A hereditary class of graphs is well-quasi-ordered by the labelled induced subgraph relation if and only if the set of prime graphs in is. In particular, is well-quasi-ordered by the labelled induced subgraph relation if and only if the set of connected graphs in is.
-free bipartite graphs are well-quasi-ordered by the labelled induced subgraph relation.
Let and be well-quasi-orders. We define the Cartesian Product of these well-quasi-orders as the order on the set where if and only if and . Lemma 3 implies that is also a well-quasi-order. If has a labelling with elements of and of , say and , we can construct the combined labelling in that labels each vertex of with the label .
Fix a well-quasi-order that has at least one element. Let be a class of graphs. For each fix a labelling . Then is well-quasi-ordered by the labelled induced subgraph relation if and only if for every well-quasi-order and every labelling of the graphs in by this order, the combined labelling in obtained from these labellings also results in a well-quasi-ordered set of labelled graphs.
If is well-quasi-ordered by the labelled induced subgraph relation then by definition it is well-quasi-ordered when labelled with labels from these combined labellings obtained from these well-quasi-orders. If is not well-quasi-ordered by the labelled induced subgraph relation then there must be a well-quasi-order and an infinite set of graphs whose vertices are labelled with elements of such that these graphs form an infinite labelled antichain. For each graph , replace the label on vertex by . The graphs are now labelled with elements of the well-quasi-order and result in an infinite labelled antichain of graphs labelled with such combined labellings. This completes the proof.∎
2.3 -uniform Graphs
For an integer , a graph is -uniform if there is a symmetric square matrix of order and a graph on vertices such that , where is the graph class defined as follows. Let be the disjoint union of infinitely many copies of . For , let be the subset of containing vertex from each copy of . Construct from an infinite graph on the same vertex set by applying a subgraph complementation to if and only if and by applying a bipartite complementation to a pair if and only if . Thus, two vertices and are adjacent in if and only if and or and . Then, is the hereditary class consisting of all the finite induced subgraphs of . The minimum such that is -uniform is the uniformicity of . The second of the next two lemmas follows directly from the above definitions.
The following result was proved by Korpelainen and Lozin. The class of disjoint unions of cliques is a counterexample for the reverse implication.
Lemma 8 ()
Any class of graphs of bounded uniformicity is well-quasi-ordered by the labelled induced subgraph relation.
The following lemma follows directly from the definition of clique-width and the definition of -uniform graphs (see also  for a more general result).
Every -uniform graph has clique-width at most .
3 Partitioning -Partite Graphs
In Section 3.1, we first introduce a graph decomposition on -partite graphs. We then show how to extend results on bounded clique-width or well-quasi-orderability by the labelled induced subgraph relation from bipartite graphs in an arbitrary hereditary class of graphs to the -partite graphs in this class that are decomposable in this way. In Section 3.2, we then give sufficient conditions for a -partite graph to have such a decomposition.
3.1 The Decomposition
Let be a -partite graph given with a partition of its vertex set into three independent sets , and . Suppose that each set can be partitioned into sets such that, taking subscripts modulo :
for if then is complete to and anti-complete to .
For let . We say that the graphs are the slices of . If the slices belong to some graph class , then we say that can be partitioned into slices from ; see Figure 2 for an example.
If is a -partite graph that can be partitioned into slices of clique-width at most then has clique-width at most .
Since every slice of has clique-width at most , it can be constructed using the labels . Applying relabelling operations if necessary, we may assume that at the end of this construction, every vertex receives the label . We can modify this construction so that we use the labels instead, in such a way that at all points in the construction, for each every constructed vertex in has a label in . To do this we replace:
creation operations by if ,
relabel operations by and
join operations by
This modified construction uses labels and at the end of it, every vertex in is labelled with label . We may do this for every slice of independently. We now show how to use these constructed slices to construct using six labels in such a way that every vertex in is labelled with label . We do this by induction. If then , so we are done. If then by the induction hypothesis, we can construct in this way. Consider the copy of constructed earlier and relabel its vertices using the operations , and so that in this copy of , every vertex in is labelled . Next take the disjoint union of the obtained graph with the constructed . Then, apply join operations , and . Finally, apply the relabelling operations , and . This constructs in such a way that every vertex in is labelled with . By induction, it follows that has clique-width at most .∎
Let be a hereditary graph class containing a class . Let be the set of -partite graphs in that can be partitioned into slices from . If is well-quasi-ordered by the labelled induced subgraph relation then so is .
For each graph in , we may fix a partition into independent sets with respect to which the graph can be partitioned into slices from . Let be the well-quasi-order with in which every pair of distinct elements is incomparable. By Lemma 7, we need only consider labellings of graphs in of the form where and belongs to an arbitrary well-quasi-order . Suppose can be partitioned into slices , with vertices labelled as in . The slices along with the labellings completely describe the edges in . Suppose is another such graph, partitioned into slices . If is smaller than under the subsequence relation, then is an induced subgraph of . The result follows by Lemma 3.∎
3.2 Curious Graphs
Let be a -partite graph given together with a partition of its vertex set into three independent sets , and . An induced or in is rainbow if it has exactly one vertex in each set . We say that is curious with respect to the partition if it contains no rainbow or when its vertex set is partitioned in this way. We say that is curious if there is a partition with respect to which is curious. In this section we will prove that given a hereditary class , if the bipartite graphs in are well-quasi-ordered by the labelled induced subgraph relation or have bounded clique-width, then the same is true for the curious graphs in .
A linear order of the vertices of an independent set is
increasing if implies ,
decreasing if implies ,
monotone if it is either increasing or decreasing.
Bipartite graphs that are -free are also known as bipartite chain graphs. It is and it is well known (and easy to verify) that a bipartite graph is -free if and only if the vertices in each independent set of the bipartition admit a monotone ordering. Suppose is a curious graph with respect to some partition . We say that (with respect to this partition) the graph is a curious graph of type if exactly of the graphs , and contain an induced .
3.2.1 Curious Graphs of Type and .
Note that if is a curious graph of type or with respect to the partition then without loss of generality, we may assume that and are both -free.
Let be a curious graph with respect to , such that and are both -free. Then the vertices of admit a linear ordering which is decreasing in and increasing in .
For a set , we use to denote the set of vertices in that are adjacent to . We may choose a linear order of the vertices of according to their neighbourhood in , breaking ties according to their neighbourhood in i.e. an order such that:
if then and
if and then .
Clearly such an ordering is decreasing in .
Suppose, for contradiction, that this order is not increasing in . Then there must be indices such that . Then by Property (ii). By Property (i) it follows that . This means that there are vertices and . Now if is adjacent to then is a rainbow and if is non-adjacent to then is a rainbow . This contradiction implies that the order is indeed decreasing in , which completes the proof.∎
If is a curious graph of type or with respect to a partition then can be partitioned into slices that are bipartite.
Let be a linear order on satisfying Lemma 12. Let and for , let . We partition and as follows. For , let . For , let . In particular, note that the vertices of and are complete and anti-complete to , respectively. The following properties hold:
If then is complete to and anti-complete to .
If then is anti-complete to and complete to .
If and is non-adjacent to then is a rainbow , a contradiction. If and is adjacent to then is a rainbow , a contradiction. It follows that:
If then is complete to .
If then is anti-complete to .
For , let . The above properties about the edges between the sets show that can be partitioned into the slices . Now, for each , is anti-complete to , so every slice is bipartite. This completes the proof.∎
3.2.2 Curious Graphs of Type and .
Fix . If is a curious graph of type with respect to a partition then can be partitioned into slices of type at most .
Fix and let be a curious graph of type with respect to a partition . We may assume that contains an induced .
Claim 1. Given a in , every vertex of has exactly two neighbours in the and these neighbours either both lie in or both lie in .
Let and induce a in such that is adjacent to but not to and is adjacent to but not to . Consider . For , if is adjacent to both and , then is a rainbow , a contradiction. Therefore can have at most one neighbour in and at most one neighbour in . If is non-adjacent to and then is a rainbow , a contradiction. Therefore, can have at most one non-neighbour in , and similarly have have at most one non-neighbour in . Therefore, if is adjacent to , then it must be non-adjacent to , so it must be adjacent to , so it must be non-adjacent to . Similarly, if is non-adjacent to then it must be adjacent to , so it must be non-adjacent to , so it must be adjacent to . Hence Claim 3.2.2 follows.
Consider a maximal set of vertex-disjoint sets that induce copies of in . We say that a vertex of distinguishes two graphs and if its neighbours in and do not belong to the same set . We group these sets into blocks that are not distinguished by any vertex of . In other words, every contains at least one and every vertex of is complete to one of the sets and and anti-complete to the other. For , let . We define a relation on the blocks as follows:
holds if is complete to , while is anti-complete to .
For distinct blocks , at most one of and can hold.
Claim 2. Let and be distinct blocks.
There is a vertex that differentiates and .
If is complete to and anti-complete to then (see also Figure 3).
If is complete to and anti-complete to then .
By definition of the blocks and there must be a vertex that distinguishes them. Without loss of generality we may assume that is complete to and anti-complete to . It remains to show that . If is adjacent to then is a rainbow , a contradiction. If is non-adjacent to then is a rainbow , a contradiction. Therefore is anti-complete to and is complete to . It follows that . Claim 3.2.2 follows by symmetry.
Claim 3. The relation is transitive.
Suppose that and . Since and are distinct, there must be a vertex that distinguishes them and combining Claim 3.2.2 with the fact that it follows that must be complete to . Suppose and . Then is adjacent to , since and is adjacent to by choice of . Now is must be non-adjacent to otherwise would be a rainbow . Therefore is anti-complete to , so distinguishes and . By Claim 3.2.2 it follows that . This completes the proof of Claim 3.2.2.
Claim 4. The set of blocks admits a linear order such that
if then is complete to , while is anti-complete to and
for each there exists an such that if then is complete to and anti-complete to and if then is anti-complete to and complete to .
Next consider the set of vertices in that do not belong to any set . Let denote this set and note that is -free by maximality of the set . For let