Many decision problems are known to be NP-complete [GJ79], and it is generally believed that such problems cannot be solved in time polynomial in the input size. For many of these hard problems, placing restrictions on the input (that is, insisting that the input has certain stated properties) can lead to significant changes in the computational complexity of the problem. This leads one to ask fundamental questions: under which input restrictions can an NP-complete problem be solved in polynomial time, and under which input restrictions does the problem remain NP-complete? For problems defined on graphs, we can restrict the input to some special class of graphs that have some commonality. The ultimate goal is to obtain complexity dichotomies for large families of graph problems, which tell us exactly for which graph classes a certain problem is efficiently solvable and for which it stays computationally hard. Such dichotomies may not always exist if P NP [La75], but rather than solving problems one by one, and graph class by graph class, we want to discover general properties of graph classes from which we can determine the tractability or hardness of families of problems.
1.1 Width Parameters
One way to define a graph class is to use a notion of “width” and consider the set of graphs for which the width is bounded by a constant. Though it will not be our focus, let us briefly illustrate this idea with the most well-known width parameter, treewidth. A tree decomposition of a graph is a tree whose nodes are subsets of and has the properties that, for each in , the tree nodes that contain induce a non-empty connected subgraph, and, for each edge in , there is at least one tree node that contains and . See Figure 1 for an illustration of a graph and one of its tree decompositions.
The sets of vertices that form the nodes of the tree are called bags and the width of the decomposition is one less than the size of the largest bag. The treewidth of is the minimum width of its tree decompositions. One can therefore define a class of graphs of bounded treewidth; that is, for some constant , the collection of graphs that each have treewidth at most . The example in Figure 1 has treewidth . Moreover, it is easy to see that trees form exactly the class of graphs with treewidth . Hence, the treewidth of a graph can be seen as a measure that indicates how close a graph is to being a tree. Many graph problems can be solved in polynomial time on trees. For such problems it is natural to investigate whether restricting the problem to inputs that have bounded treewidth still yields algorithmic tractability. An approach that often yields polynomial-time algorithms is to apply dynamic programming over the decomposition tree. A disadvantage of this approach is that only sufficiently sparse graphs have bounded treewidth.
We further discuss reasons for focussing on width parameters in Section 1.2, but let us first note that there are many alternative width parameters, each of which has led to progress in understanding the complexity of problems on graphs.
Clique-width, the central width parameter in our survey, is another well-known example, which has received significant attention since it was introduced by Courcelle, Engelfriet and Rozenberg [CER93] at the start of the 1990s. Clique-width can be seen as a generalisation of treewidth that can deal with dense graphs, such as complete graphs and complete bipartite graphs, provided these instances are sufficiently regular. We will give explain this in Section 3, where we also give a formal definition, but, in outline, the idea is, given a graph , to determine how it can be built up vertex-by-vertex using four specific graph operations that involve assigning labels to the vertices. The operations ensure that vertices labelled alike will keep the same label and thus, in some sense, behave identically. The clique-width of is the minimum number of different labels needed to construct in this way. Hence, if the clique-width of a graph is small, we can decompose into large sets of similarly behaving vertices, and these decompositions can be exploited to find polynomial-time algorithms (as we shall see later in this paper).
We remark that many other width parameters have been defined including boolean-width, branch-width, MIM-width, MM-width, module-width, NLC-width, path-width and rank-width, to name just a few. These parameters differ in strength, as we explain below; we refer to [Gu17, HOSG08, KLM09, Va12] for surveys on width parameters.
Given two width parameters and , we say that dominates if there is a function such that for all graphs . If dominates but not the reverse, then is more general than , as is bounded for larger graph classes: whenever is bounded for some graph class, then this is also the case for , but there exists an infinite family of graphs for which the reverse does not hold. If dominates and dominates , then and are equivalent. For instance, MIM-width is more general than boolean-width, clique-width, module-width, NLC-width and rank-width, all of which are equivalent [BTV11, Jo98, OS06, Ra08, Va12]. The latter parameters are more general than the equivalent group of parameters branch-width, MM-width and treewidth, which are, in turn, more general than path-width [CO00, RS91, Va12]. To give a concrete example, recall that the treewidth of the class of complete graphs is unbounded, in contrast to the clique-width. More precisely, a complete graph on vertices has treewidth but clique-width . As another example, the reason that rank-width and clique-width are equivalent is because the inequalities hold for every graph [OS06]. These two inequalities are essentially tight [Ou17], and, as such, the latter example also shows that two equivalent parameters may not necessarily be linearly, or even polynomially, related.
1.2 Motivation for Width Parameters
The main computational reason for the large interest in width parameters is that many well-known NP-complete graph problems become polynomial-time solvable if some width parameter is bounded. There are a number of meta-theorems which prescribe general, sufficient conditions for a problem to be tractable on a graph class of bounded width. For treewidth and equivalent parameters, such as branch-width and MM-width, one can use the celebrated theorem of Courcelle [Co90]. This theorem, slightly extended from its original form, states that for every graph class of bounded treewidth, every problem definable in MSO can be solved in time linear in the number of vertices of the graph.111MSO refers to the fragment of second order logic where quantified relation symbols must have arity at most , which means that, with graphs, one can quantify over both sets of vertices and sets of edges. Many graph problems can be defined using MSO, such as deciding whether a graph has a -colouring (for fixed ) or a Hamiltonian path, but there are also problems that cannot be defined in this way. In order to use this theorem, one can use the linear-time algorithm of Bodlaender [Bo96] to verify whether a graph has treewidth at most for any fixed constant (that is, is not part of the input). However, many natural graph classes, such as all those that contain graphs with arbitrarily large cliques, have unbounded treewidth.
We have noted that clique-width is more general than treewidth. This means that if we have shown that a problem can be solved in polynomial time on graphs of bounded clique-width, then it can also be solved in polynomial time on graphs of bounded treewidth. Similarly, if a problem is NP-complete for graphs of bounded treewidth, then the same holds for graphs of bounded clique-width. For graph classes of bounded clique-width, one can use several other meta-theorems. The first such result is due to Courcelle, Makowsky and Rotics [CMR00]. They proved that graph problems that can be defined in MSO are linear-time solvable on graph classes of bounded clique-width.222MSO is monadic second order logic with the use of quantifiers permitted on relations of arity (such as vertices), but not of arity (such as edges) or more. Hence, MSO is more restricted than MSO. We refer to [CE12] for more information on MSO and MSO. An example of such a problem is the well-known Dominating Set problem. This problem is to decide, for a graph and integer , if contains a set of size at most such that every vertex of has at least one neighbour in .333Several other problems, such as List Colouring and Precolouring Extension are polynomial-time solvable on graphs of bounded treewidth [JS97], but stay NP-complete on graph of bounded clique-width; the latter follows from results of [JS97] and [BDM09], respectively; see also [GJPS17].
1.3 Focus: Clique-Width
As mentioned, in this survey we focus on clique-width. Despite the usefulness of boundedness of clique-width, our understanding of clique-width itself is still very limited. For example, although computing the clique-width of a graph is known to be NP-hard in general [FRRS09],444It is also NP-hard to compute treewidth [ACP87] and parameters equivalent to clique-width, such as NLC-width [GW05], rank-width (see [HO08, Oum08]) and boolean-width [SV16]. the complexity of computing the clique-width is open even on very restricted graph classes, such as unit interval graphs (see [HMR10] for some partial results). To give another example, the complexity of determining whether a given graph has clique-width at most is still open for every fixed constant . On the positive side, see [CHLRR12] for a polynomial-time algorithm for and [EGW03] for a polynomial-time algorithm, for every fixed , on graphs of bounded treewidth.
To get a better handle on clique-width, many properties of clique-width, and relationships between clique-width and other graph parameters, have been determined over the years. In particular, numerous graph classes of bounded and unbounded clique-width have been identified. This has led to several dichotomies for various families of graph classes, which state exactly which graph classes of the family have bounded or unbounded clique-width. However, determining (un)boundedness of clique-width of a graph class is usually a highly non-trivial task, as it requires a thorough understanding of the structure of graphs in the class. As such, there are still many gaps in our knowledge.
A number of results on clique-width are collected in the surveys on clique-width by Gurski [Gu17] and Kamiński, Lozin and Milanič [KLM09]. Gurski focuses on the behaviour of clique-width (and NLC-width) under graph operations and graph transformations. Kamiński, Lozin and Milanič also discuss results for special graph classes. We refer to a recent survey of Oum [Ou17] for algorithmic and structural results on the equivalent width parameter rank-width.
1.4 Aims and Outline
In Section 2 we introduce some basic terminology and notation that we use throughout the paper. In Section 3 we formally define clique-width. In the same section we present a number of basic results on clique-width and explain two general techniques for showing that the clique-width of a graph class is bounded or unbounded. For this purpose, in the same section we also list a number of graph operations that preserve (un)boundedness of clique-width for hereditary graph classes.
A graph class is hereditary if it is closed under taking induced subgraphs, or equivalently, under vertex deletion. Due to its natural definition, the framework of hereditary graph classes captures many well-known graph classes, such as bipartite, chordal, planar, interval and perfect graphs; we refer to the textbook of Brandstädt, Le and Spinrad [BLS99] for a survey. As we shall see, boundedness of clique-width has been particularly well studied for hereditary graph classes. We discuss the state-of-the-art and other known results on boundedness of clique-width for hereditary graph classes in Section 4. This is all related to our first aim: to update the paper of Kamiński, Lozin and Milanič [KLM09] from 2009 by surveying, in a systematic way, known results and open problems on boundedness of clique-width for hereditary graph classes.
Our second aim is to discuss algorithmic implications of the results from Section 4. We do this in Section 5 by focussing on two well-known problems. We first discuss implications for the Colouring problem, which is well known to be NP-complete [Lo73]. We focus on (hereditary) graph classes defined by two forbidden induced subgraphs. Afterwards, we consider the algorithmic consequences for the Graph Isomorphism problem. This problem can be solved in quasi-polynomial time [Ba16]. It is not known if Graph Isomorphism can be solved in polynomial time, but it is not NP-complete unless the polynomial hierarchy collapses [Sc88]. As such, we define the complexity class GI, which consists of all problems that can be polynomially reduced to Graph Isomorphism and a problem in GI is GI-complete if Graph Isomorphism can be polynomially reduced to it. The Graph Isomorphism problem is of particular interest, as there are similarities between proving unboundedness of clique-width of some graph class and proving that Graph Isomorphism stays GI-complete on this class [Sc17].
Our third aim is to discuss a conjectured relationship between boundedness of clique-width and well-quasi-orderability by the induced subgraph relation. If it can be shown that a graph class is well-quasi-ordered, we can apply several powerful results to prove further properties of the class. This is, for instance, illustrated by the Robertson-Seymour Theorem [RS04-Wagner], which states that the set of all finite graphs is well-quasi-ordered by the minor relation. This result makes it possible to test in cubic time whether a graph belongs to some given minor-closed graph class [RS95] (see [KKR12] for a quadratic algorithm). For the induced subgraph relation, it is easy to construct examples of hereditary graph classes that are not well-quasi-ordered. Take, for instance, the class of graphs of degree at most , which contains an infinite anti-chain, namely the set of all cycles.
If every hereditary graph class that is well-quasi-ordered by the induced subgraph relation also has bounded clique-width, then all algorithmic consequences of having bounded clique-width would also hold for being well-quasi-ordered by the induced subgraph relation. However, Lozin, Razgon and Zamaraev [LRZ18] gave a negative answer to a question of Daligault, Rao and Thomassé [DRT10] about this implication, by presenting a hereditary graph class of unbounded clique-width that is nevertheless well-quasi-ordered by the induced subgraph relation. Their graph class can be characterized only by infinitely many forbidden induced subgraphs. This led the authors of [LRZ18] to conjecture that every finitely defined hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width, which, if true, would still be very useful. All known results agree with this conjecture, and we survey these results in Section 6. In the same section we explain that the graph operations given in Section 3 do not preserve well-quasi-orderability by the induced subgraph relation. However, we also explain that a number of these operations can be used for a stronger property, namely well-quasi-orderability by the labelled induced subgraph relation.
In Section 7 we conclude our survey with a list of other relevant open problems. There, we also discuss some variants of clique-width, including linear clique-width and power-bounded clique-width.
Throughout the paper we consider only finite, undirected graphs without multiple edges or self-loops.
Let be a graph. The degree of a vertex is the size of its neighbourhood . For a subset , the graph denotes the subgraph of induced by , which is the graph with vertex set and an edge between two vertices if and only if . If is an induced subgraph of , then we denote this by . Note that can be obtained from by deleting the vertices of . The line graph of is the graph with vertex set and an edge between two vertices and if and only if and share a common end-vertex in .
An isomorphism from a graph to a graph is a bijective mapping such that there is an edge between two vertices and in if and only if there is an edge between and in . If such an isomorphism exists then and are said to be isomorphic. We say that is -free if contains no induced subgraph isomorphic to .
Let be a graph. A set is a clique of and is complete if there is an edge between every pair of vertices in . If is connected, then a vertex is a cut-vertex of if is disconnected, and a clique is a clique cut-set of if is disconnected. If is connected and has at least three vertices but no cut-vertices, then is -connected. A maximal induced subgraph of that has no cut-vertices is a block of . If is connected and has no clique cut-set, then is an atom.
The graphs , and denote the cycle, path and complete graph on vertices, respectively. The length of a path or a cycle is the number of its edges. The distance between two vertices and in a graph is the length of a shortest path between them. For an integer , the -th power of is the graph with vertex set and an edge between two vertices and if and only if and are at distance at most from each other in .
If and are graphs with disjoint vertex sets, then the disjoint union of and is the graph . The disjoint union of copies of a graph is denoted . A forest is a graph with no cycles, that is, every connected component is a tree. A forest is linear if it has no vertices of degree at least , or equivalently, if it is the disjoint union of paths. A leaf in a tree is a vertex of degree . In a complete binary tree all non-leaf vertices have degree .
Let and be disjoint vertex subsets of a graph . A vertex is (anti-)complete to if it is (non-)adjacent to every vertex in . Similarly, is (anti-)complete to if every vertex in is (non-)adjacent to every vertex in . A set of vertices is a module of if every vertex of that is 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. We say that is prime if every module of is trivial.
A graph is bipartite if its vertex set can be partitioned into two (possibly empty) subsets and such that every edge of has one end-vertex in and the other one in . If is complete to , then is complete bipartite. For two non-negative integers and , we denote the complete bipartite graph with partition classes of size and , respectively, by . The graph is also known as the -vertex star. The subdivision of an edge in a graph replaces by a new vertex and edges and . We let and be the graphs obtained from by subdividing one of its edges once or twice, respectively (see Figure 2 for examples).
|( shown)||( shown)|
A graph is complete -partite, for some , if its vertex set can be partitioned into independent sets such that there exists an edge between two vertices and if and only if and do not belong to the same set . Note that a non-empty graph is complete -partite for some if and only if it is -free.
Let be a graph. Its complement is the graph with vertex set and an edge between two vertices and if and only if is not an edge of . We say that is self-complementary if is isomorphic to . The complement of a bipartite graph is a co-bipartite graph.
|( shown)||( shown)||( shown)||( shown)|
The graphs , , , and are also known as the claw, diamond, gem, and house, respectively. The latter three graphs are shown in Figure 3, along with the domino. The graph , for , denotes the subdivided claw, which is the tree with one vertex of degree and exactly three leaves, which are of distance , and from , respectively. Note that , and . See Figure 3 for an example. We let be the class of graphs every connected component of which is either a subdivided claw or a path on at least one vertex. The graph with denotes the triangle with pendant paths of length , and , respectively. That is, is the graph with vertices , and and edges , , , for , for and for . Note that . The graphs , , and are also known as the paw, bull, net and hammer, respectively; see also Figure 4. Also note that is the line graph of . We let be the class of graphs that are the line graphs of graphs in . Note that contains every graph and every path (as the line graph of is for ).
Let be a graph. For an induced subgraph , the subgraph complementation operation, which acts on with respect to , replaces every edge in by a non-edge, and vice versa. If we apply this operation on with respect to itself, then we obtain the complement of . For two disjoint vertex subsets and in , the bipartite complementation operation, which acts on with respect to and , replaces every edge with one end-vertex in and the other one in by a non-edge and vice versa. We note that applying a bipartite complementation is equivalent to applying a sequence of three consecutive subgraph complementations, namely on , and .
Let be a graph class. Denote the number of labelled graphs on vertices in by . Then is superfactorial if there does not exist a constant such that for every .
Recall that a graph class is hereditary if it is closed under taking induced subgraphs. It is not difficult to see that a graph class is hereditary if and only if can be characterized by a unique set of minimal forbidden induced subgraphs. A hereditary graph class is finitely defined if is finite. We note, however, that the set may have infinite size. For example, if is the class of bipartite graphs, then . If is a set of graphs, we say that a graph is -free if does not contain any graph in as an induced subgraph. In particular, this means that if a graph class is hereditary, then is exactly the class of -free graphs. If or for some , we may also describe a graph as being -free or -free, respectively, rather than -free; recall that if we may write -free instead.
Let and be sets of graphs. The class of -free graphs is contained in the class of -free graphs if and only if for every graph , the set contains an induced subgraph of .
Suppose and are sets of graphs such that for every graph , the set contains an induced subgraph of . Observation 2 implies that any graph problem that is polynomial-time solvable for -free graphs is also polynomial-time solvable for -free graphs, and any graph problem that is NP-complete for -free graphs is also NP-complete for -free graphs.
We define the complement of a hereditary graph class as . Then is closed under complementation if . As is the unique minimal set of forbidden induced subgraphs for , we can make the following observation.
A hereditary graph class is closed under complementation if and only if is closed under complementation.
Let be a graph. The contraction of an edge replaces and and their incident edges by a new vertex and edges if and only if either or was an edge in (without creating multiple edges or self-loops). Let be a vertex with exactly two neighbours , which in addition are non-adjacent. The vertex dissolution of removes , and , and adds the edge . Note that vertex dissolution is a special type of edge contraction, and it is the reverse operation of an edge subdivision (recall that the latter operation replaces an edge by a new vertex with edges and ).
Let and be graphs. The graph is a subgraph of if can be modified into by a sequence of vertex deletions and edge deletions. We can define other containment relations using the graph operations defined above. We say that contains as a minor if can be modified into by a sequence of edge contractions, edge deletions and vertex deletions, as a topological minor if can be modified into by a sequence of vertex dissolutions, edge deletions and vertex deletions, as an induced minor if can be modified into by a sequence of edge contractions and vertex deletions, and as an induced topological minor if can be modified into by a sequence of vertex dissolutions and vertex deletions. Let be a set of graphs. If does not contain any of the graphs as a subgraph, then is -subgraph-free. We define the terms -minor-free, -topological-minor-free, -induced-minor-free, and -induced-topological-minor-free analogously. Note that graph classes defined by some set of forbidden subgraphs, minors, topological minors, induced minors, or induced topological minors are hereditary, as they are all closed under vertex deletion.
A graph is planar if it can be embedded in the plane in such a way that any two edges only intersect with each other at their end-vertices. It is well known that the class of planar graphs can be characterized by a set of forbidden minors: Wagner’s Theorem [Wa37] states that a graph is planar if and only if it is -minor-free.
We will also need the following folklore observation (see, for example, [GPR15]).
For every , a graph is -subgraph-free if and only if it is -minor-free.
A -colouring of a graph is a mapping such that whenever and are adjacent vertices. The chromatic number of is the smallest such that has a -colouring. The clique number of is the size of a largest clique of .
A graph is perfect if, for every , the chromatic number of is equal to the clique number of . The Strong Perfect Graph Theorem [CRST06] states that is perfect if and only if is -free and -free. A graph is chordal if it is -free and weakly chordal if it is -free and -free. A graph is a split graph if it has a split partition, that is, a partition of its vertex set into two (possibly empty) sets and , where is a clique and is an independent set. It is known that a graph is split if and only if it is -free [FH77]. A graph is a permutation graph if line segments connecting two parallel lines can be associated to its vertices in such a way that two vertices of are adjacent if and only if the two corresponding line segments intersect. A graph is a permutation split graph if it is both permutation and split, and is a permutation bipartite graph if it is both permutation and bipartite. A graph is chordal bipartite if it is -free. A graph is distance-hereditary if the distance between any two vertices and in any connected induced subgraph of is the same as the distance of and in . Equivalently, a graph is distance-hereditary if and only if it is -free [BM86]. A graph is (unit) interval if it has a representation in which each vertex corresponds to an interval (of unit length) of the line such that two vertices and are adjacent if and only if .
We make the following observation. A number of inclusions in Observation 2 follow immediately from the definitions and the Strong Perfect Graph Theorem. For the remaining inclusions we refer to [BLS99].
The following statements hold:
every split graph is chordal
every (unit) interval graph is chordal
every chordal graph is weakly chordal
every (bipartite or split) permutation graph is weakly chordal
every distance-hereditary graph is weakly chordal
every weakly chordal graph is perfect
every bipartite permutation graph is chordal bipartite, and
every (chordal) bipartite graph is perfect.
The containments listed in Observation 2 (and those that follow from them by transitivity) are also displayed Figure 5. It is not difficult to construct counterexamples for the other containments. Indeed, for pairs of classes above for which we have listed the minimal forbidden induced subgraph characterizations, these characterizations immediately provide such counterexamples.
We now introduce the notion of treewidth formally. Recall from Section 1 that treewidth expresses to what extent a graph is “tree-like”. A tree decomposition of a graph is a pair where is a tree and is a collection of subsets of , such that the following three conditions hold:
for every edge , there is an such that and
for every , the set induces a connected subtree of .
The width of the tree decomposition is , and the treewidth of is the minimum width over all tree decompositions of . If is a path, then is a path decomposition of . The path-width of is the minimum width over all path decompositions of .
A quasi order on a set is a reflexive, transitive binary relation. Two elements in are comparable if or ; otherwise they are incomparable. A set of pairwise (in)comparable elements in is called an (anti)-chain. A quasi-order is a well-quasi-order if every infinite sequence of elements in contains a pair with and , or equivalently, if has no infinite strictly decreasing sequence and no infinite anti-chain. A partial order is a quasi-order which is anti-symmetric, that is, if and then . If we consider two graphs to be “equal” when they are isomorphic, then all quasi orders considered in this paper are in fact partial orders. As such, throughout this paper “quasi order” can be interpreted as “partial order”.
For an arbitrary set , we 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.
The following lemma is well known and very useful when dealing with quasi-orders.
[Higman’s Lemma [Higman52]] Let be a well-quasi-order. Then is a well-quasi-order.
In this section we give a number of basic results on clique-width. We begin by giving a formal definition.555The term clique-width and the definition in essentially same form we give here were introduced by Courcelle and Olariu [CO00] based on operations and related decompositions from Courcelle, Engelfriet and Rozenberg [CER93]; see also [CE12]. Although we consider only undirected graphs, the definitions of [CO00] also covered the case of directed graphs. Other equivalent width parameters have also been studied for directed graphs. For example, Kanté and Rao [KR13] considered the rank-width of directed graphs. The clique-width of a graph , denoted by , is the minimum number of labels needed to construct using the following four operations:
Create a new graph with a single vertex with label . (This operation is written .)
Take the disjoint union of two labelled graphs and (written ).
Add an edge between every vertex with label and every vertex with label , (written ).
Relabel every vertex with label to have label (written ).
We say that a construction of a graph with the four operations is a -expression if it uses at most labels. Thus the clique-width of is the minimum for which has a -expression. We refer to [CHMPR15, HMPR16, HS15] for a number of characterizations of clique-width and to [Ka18] for a compact representation of graphs of clique-width .
We first note that and . Now consider a path on four vertices , in that order. Then this path can be constructed using the four operations (using only three labels) as follows:
Note that at the end of this construction, only has label . It is easy to see that a construction using only two labels is not possible. Hence, we deduce that . This construction can readily be generalized to longer paths: for let be a -expression for the path on vertices , with only the vertex having label , then is a -expression for the path on vertices , with only the vertex having label . Therefore for all . Moreover, by changing the construction to give the first vertex on a path a unique fourth label, we can connect it to the last constructed vertex of (the only vertex with label ) via an edge-adding operation to obtain . Hence, we find that for every . In fact holds for every [MR99].
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 . If such a constant does not exist, we say that the clique-width of is unbounded. A hereditary graph class is a minimal class of unbounded clique-width if it has unbounded clique-width and every proper hereditary subclass of has bounded clique-width.
The following two observations, which are both well known and readily seen, give two graph classes of small clique-width. In particular, Proposition 3 follows from Example 3 after observing that a graph of maximum degree at most is the disjoint union of paths and cycles. For more examples of graph classes of small width, see, for instance, [BDLM05, BELL06].
Every forest has clique-width at most .
Let be a tree with a root vertex . We claim that there is a -expression which creates such that, in the resulting labelled tree, only has label . We prove this by induction on . Clearly this holds when . Otherwise, let be the children of and let be the subtrees of rooted at , respectively. By the induction hypothesis, for each there is a -expression which creates such that, in the resulting labelled tree, only has label . We take the disjoint union of these expressions and let be the resulting -expression. Then is a -expression which creates such that, in the resulting labelled tree, only has label . Therefore for every tree , there is a -expression that constructs . Since a forest is a disjoint union of trees, we can then use the operation to extend this to a -expression for any forest. The proposition follows. ∎
Every graph of maximum degree at most has clique-width at most .
Recall that for general graphs, the complexity of computing the clique-width of a graph was open for a number of years, until Fellows, Rosamund, Rotics and Szeider [FRRS09] proved that this is NP-hard. However, Proposition 3 implies that we can determine the clique-width of a forest in polynomial time: if contains an induced , then ; if is -free but has an edge, then ; and if for some , then .
In contrast to Proposition 3, graphs of maximum degree at most may have arbitrarily large clique-width. An example of this is a wall of arbitrary height, which can be thought of as a hexagonal grid. We do not formally define the wall, but instead we refer to Figure 6, in which three examples of walls of different heights are depicted; see, for example, [Ch15] for a formal definition. Note that walls of height at least have maximum degree . The following result is well known; see for example [KLM09].
The class of walls has unbounded clique-width.
As mentioned, clique-width is more general than treewidth. Courcelle and Olariu [CO00] proved that for every graph (see [GM03] for an alternative proof). Corneil and Rotics [CR05] improved this bound by showing that for every graph . They also proved that for every , there is a graph with and . The following result shows that for restricted graph classes the two parameters may be equivalent (see [Co18, Co18b] for graph classes for which treewidth and clique-width are even linearly related).
[[GW00]] For , every class of -subgraph-free graphs of bounded clique-width has bounded treewidth.
A class of graphs of bounded maximum degree has bounded clique-width if and only if it has bounded treewidth.
Gurski and Wanke gave another connection between treewidth and clique-width.
[[GW07]] A class of graphs has bounded treewidth if and only if the class of line graphs of graphs in has bounded clique-width.
As mentioned in Section 1, boundedness of clique-width has been determined for many hereditary graph classes. However, using the definition of clique-width directly to prove that a certain hereditary graph class has bounded clique-width is often difficult. An alternative way to show that a hereditary graph class has bounded clique-width is to prove that for infinitely many values of , the number of labelled graphs in on vertices is at most the Bell number [ALR09], but this has limited applicability. The following BCW Method is more commonly used:
Bounding Clique-Width (BCW Method)
If possible, consider only graphs in that have some suitable property .
Take a graph class for which it is known that its clique-width is bounded.
For every graph (possibly with property ), reduce to a graph in by using a constant number of graph operations that do not change the clique-width of by “too much”.
Note that the subclass of graphs in that have some property in Step 1 need not be hereditary. For example, it is known [BL02, LR04] that we may choose the property to be that of being -connected and that we can delete some constant number of vertices from a graph without affecting the clique-width by more than some bounded amount. Then we could try to prove that has bounded clique-width by showing that for every -connected graph in , we can delete no more than vertices to obtain a graph in some class that we know to have bounded clique-width. We give some concrete examples of this method in the next section.
The power of the method depends on both the graph property in Step 1 and the graph operations that we are allowed to use in Step 3. In particular we will use graph operations to modify a graph of some class into the disjoint union of some graphs that have a simpler structure than itself. As a result, we can then deal with these simpler graphs separately. This approach is particularly useful if is hereditary: if the simpler graphs are induced subgraphs of the original graph , then we can still make use of earlier deduced properties for when dealing with the simpler induced subgraphs of . Before giving important examples of these operations and properties, we first formalize our approach.
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 can be obtained from a graph in by performing at most times, and
for every graph there exists at least one graph in that can be obtained from by performing at most times (note that is not necessarily a subclass of ).
A graph operation preserves boundedness of clique-width if, for every finite constant and every graph class , every graph class that is -obtained from has bounded clique-width if and only if has bounded clique-width. We note that Condition 1 is necessary for this definition to be meaningful; without this condition the class of all graphs (which has unbounded clique-width) would be -obtained from every other graph class. Similarly, we also need Condition 2, as otherwise every graph class would be -obtained from the class of all graphs. If is allowed, then preserves boundedness of clique-width ad infinitum. Similarly, a graph property preserves boundedness of clique-width if, for every graph class , the subclass of with property has bounded clique-width if and only if has bounded clique-width. If necessary, we may restrict these definitions to only be valid for some specific types of graph classes.
We refer to the survey of Gurski [Gu17] for a detailed overview of graph operations that preserve boundedness of clique-width and for bounds that tell us more precisely by how much the clique-width can change when applying various operations.666We note that some of these graph operations may exponentially increase the upper bound of the clique-width. Here, we only state the most important graph operations, together with two well-known properties that preserve boundedness of clique-width.
Facts about clique-width:
Vertex deletion preserves boundedness of clique-width [LR04].
Subgraph complementation preserves boundedness of clique-width [KLM09].
Bipartite complementation preserves boundedness of clique-width [KLM09].
Being prime preserves boundedness of clique-width for hereditary graph classes [CO00].
Being -connected preserves boundedness of clique-width for hereditary graph classes [BL02, LR04].
Edge subdivision preserves boundedness of clique-width ad infinitum for graph classes of bounded maximum degree [KLM09].
We note that Fact 3 follows from Fact 2, as bipartite complementations can be mimicked by three subgraph complementations. Moreover, an edge deletion is a special case of subgraph complementation, whereas an edge contraction is a vertex deletion and a bipartite complementation. Finally, recall that an edge subdivision is the reverse operation of a vertex dissolution, which can be seen as a type of edge contraction. Hence, from Facts 1–3 it follows that edge deletion, edge contraction and edge subdivision each preserve boundedness of clique-width.
Vertex deletions, edge deletions and edge contractions do not preserve boundedness of clique-width ad infinitum: one can take any graph class of unbounded clique-width and apply one of these operations until one obtains the empty graph or an edgeless graph. Hence, Facts 1–3 do not preserve boundedness of clique-width ad infinitum. This holds even for graphs of maximum degree at most , as the class of walls and their induced subgraphs has unbounded clique-width by Theorem 3.
In contrast, Fact 6 says that edge subdivisions applied on graphs of bounded maximum degree do preserve boundedness of clique-width ad infinitum. We note that Fact 6 follows from Corollary 3 and the fact that an edge subdivision does not change the treewidth of a graph (see, for example, [LR06]). However, the condition on the maximum degree is necessary for the “only if” direction of Fact 6. Otherwise, as discussed in [DP16], one could start with a clique on at least two vertices (which has clique-width ) and then apply an edge subdivision on an edge in if and only if is not an edge in some graph of arbitrarily large clique-width with . This yields a graph that contains as an induced subgraph, implying that , which is arbitrarily larger than .
As an aside, note that edge contractions do not increase the clique-width of graphs of bounded maximum degree either. We can apply Corollary 3 again after observing from the definition of treewidth that edge contractions do not increase treewidth. However, the condition on the maximum degree is necessary here as well; a (non-trivial) counterexample is given by Courcelle [Co14], who proved that the class of graphs that are obtained by edge contractions from the class of graphs of clique-width has unbounded clique-width.
For the BCW Method, operations that preserve boundedness of clique-width may be combined, but these operations may not always be used in combination with some property that preserves boundedness of clique-width. This is because applying a graph operation may result in a graph that does not have property . Moreover, it is not always clear whether two or more properties that preserve boundedness of clique-width may be unified into one property. For instance, every non-empty class of -connected graphs is not hereditary and every class of prime graphs containing a graph on more than two vertices is not hereditary. As such, it is unknown whether Facts 4 and 5, which may only be applied on hereditary graph classes, can be combined. That is, the following problem is open.
Let be a hereditary class of graphs and let be the class of -connected prime graphs in . If has bounded clique-width, does this imply that has bounded clique-width?
To prove that a graph class has unbounded clique-width, a similar method to the BCW Method can be used.
Unbounding Clique-Width (UCW Method)
Take a graph class known to have unbounded clique-width.
For every graph , reduce to a graph in by using a constant number of graph operations that do not change the clique-width of by “too much”.
By Theorem 3, we can consider the class of walls as a starting point for the graph class . A -subdivided wall is a graph obtained from a wall after subdividing each edge exactly times for some constant . Combining Fact 6 with Theorem 3 and the observation that walls of height at least have maximum degree leads to the following result.
[[LR06]] For any constant , the class of -subdivided walls has unbounded clique-width.
Corollary 3 has proven to be very useful. For instance, it can be used to obtain the following result (recall that is the class of graphs each connected component of which is either a subdivided claw or a path).
[[LR06]] Let be a finite set of graphs. If for all , then the class of -free graphs has unbounded clique-width.
As a side note, we remark that “limit classes” of hereditary graph classes of unbounded clique-width may have bounded clique-width. For instance, the class of -subgraph-free graphs has unbounded clique-width for any two integers and due to Corollary 3. However, for every , the class of -subgraph-free graphs has bounded clique-width [LM13]. We refer to [LM13] for more details on limit classes.
Corollary 3 is further generalized by the following theorem.
[[DP16]] For and the clique-width of a graph is at least if has a partition into sets with the following properties:
is connected for all
is connected for all
for , if a vertex of is adjacent to a vertex of then
for , if a vertex of is adjacent to a vertex of then , and
for , if a vertex of is adjacent to a vertex of then and .
Many other constructions of graphs of large clique-width follow from Theorem 3 using the UCW Method (possibly by applying Facts 1–3). For instance, this is the case for square grids [MR99], whose exact clique-width was determined by Golumbic and Rotics [GR00]. This is also the case for the constructions of Brandstädt, Engelfriet, Le and Lozin [BELL06], Lozin and Volz [LV08], Korpelainen, Lozin and Mayhill [KLM14] and Kwon, Pilipczuk and Siebertz [KPS17] for proving that the classes of -free co-chordal graphs, -free bipartite graphs, split permutation graphs and twisted chain graphs, respectively, have unbounded clique-width.
Constructions of graphs of arbitrarily large clique-width not covered by Theorem 3 can be found in [GR00] and [BL03], which prove that unit interval graphs and bipartite permutation graphs, respectively, have unbounded clique-width. We discuss these results in more detail in the next section, but we note the following.
First, the classes of split permutation graphs (and the analogous bipartite class of bichain graphs) [ABLS], unit interval graphs [Lo11] and bipartite permutation graphs [Lo11] are even minimal hereditary graph classes of unbounded clique-width. Collins, Foniok, Korpelainen, Lozin and Zamaraev [CFKLZ] proved that the number of minimal hereditary graphs of unbounded clique-width is infinite. Second, for classes, such as split graphs, bipartite graphs, co-bipartite graphs and -free graphs, unboundedness of clique-width also follows from the fact that these classes are superfactorial [BL02] and an application of the following result.
[[BL02]] Every superfactorial graph class has unbounded clique-width.
4 Results on Clique-Width for Hereditary Graph Classes
In this section we survey known results on (un)boundedness of clique-width for hereditary graph classes in a systematic way.777The Information System on Graph Classes and their Inclusions [isgci] also keeps a record of many graph classes for which boundedness or unboundedness of clique-width is known. The proofs of these results often use the BCW Method or UCW Method. As mentioned earlier, many well-studied graph classes are hereditary. From the point of view of clique-width, these are also natural classes to consider, as the definition of clique-width implies that if a graph contains a graph as an induced subgraph, then .
Recall that a graph class is hereditary if and only if it can be characterized by a (possibly infinite) set of forbidden induced subgraphs . We start by giving a dichotomy for the case when consists of a single graph . This result is folklore: observe that has clique-width and see [CO00] for a proof that -free graphs have clique-width at most and [DP16] for a proof of the other claims of Theorem 4.
Let be a graph. The class of -free graphs has bounded clique-width if and only if is an induced subgraph of . Furthermore, a graph has clique-width at most if and only if it is -free.
Note that by Theorem 4 we can test whether a graph has clique-width at most in polynomial time by checking whether is -free. We recall that deciding whether a graph has clique-width at most is known to be polynomial-time solvable for [CHLRR12], but open for .
As discussed in Section 1, an important reason for studying boundedness of clique-width for special graph classes is to obtain more classes of graphs for which a wide range of classical NP-complete problems become polynomial-time solvable. Theorem 4 shows that this cannot be done for (most) classes of -free graphs. In order to find more graph classes of bounded clique-width, we can follow several approaches that try to extend Theorem 4.
To give an example, Vanherpe [Va04] considered the class of partner-limited graphs, which were introduced by Roussel, Rusu and Thuillier in [RRT99]. A vertex in a graph is a partner of an induced subgraph isomorphic to of if induces at least two s in . A graph is said to be partner-limited if every induced has at most two partners. Vanherpe proved that the clique-width of partner-limited graphs is at most . This result generalized a corresponding result of Courcelle, Makowsky and Rotics [CMR00] for -tidy graphs, which are graphs in which every induced has at most one partner.
To give another example, Makowsky and Rotics [MR99] considered the classes of -graphs, which were introduced by Babel and Olariu in [BO98]. For two integers and , a graph is a -graph if every subset of vertices induces a subgraph that has at most distinct induced s. Note that -free graphs are the -graphs, whereas -graphs are also known as -sparse graphs; note that the latter class of graphs is a subclass of the class of -tidy graphs. Makowsky and Rotics proved the following result.
[[MR99]] Let and . Then the class of -graphs has bounded clique-width if
and , or
and it has unbounded clique-width if
and , or
Theorem 4 covers all cases except where and . Makowsky and Rotics [MR99] therefore posed the following open problem (see also [KLM09]).
Is the clique-width of -graphs bounded if ?
Below we list five other systematic approaches, which we discuss in detail in the remainder of this section. First, we can try to replace “-free graphs” by “-free graphs in some hereditary graph class ” in Theorem 4. We discuss this line of research in Section 4.1.
Second, we may try to determine boundedness of clique-width of hereditary graph classes for which is small.
However, even the classification for -free graphs is not straightforward and is still incomplete.
We discuss the state-of-the-art for -free graphs in Section 4.2.
There, we also explain how results in Section 4.1 are helpful for proving results for -free graphs.888 We emphasize that the underlying research goal is not to start classifying the case of three forbidden induced subgraphs
We emphasize that the underlying research goal is not to start classifying the case of three forbidden induced subgraphs, and after the classification for two graphs and has been completed. Instead the aim is to develop new techniques through a systematic study, by looking at hereditary graph classes from different angles in order to increase our understanding of clique-width.
Third, we may try to determine boundedness of clique-width for hereditary graph classes for which only contains graphs of small size. For instance, Brandstädt, Dragan, Le and Mosca [BDLM05] classified boundedness of clique-width for those hereditary graph classes for which consists of -vertex extensions of . We discuss their result, together with other results in this direction, in Section 4.3.
Fourth, we observe that is self-complementary. As such we can try to extend Theorem 4 to graph classes closed under complementation. Determining boundedness of clique-width for such graph classes is also natural to consider due to Fact 2. We present the current state-of-the-art in this direction in Section 4.4.
Fifth, we may consider hereditary graph classes that can be described not only in terms of forbidden induced subgraphs but also using some other forbidden subgraph containment. For instance, we can consider hereditary graph classes characterized by some set of forbidden minors. We survey the known results in this direction in Section 4.5.
4.1 Considering -Free Graphs Contained in Some Hereditary Graph Class
Theorem 4 shows that the class of -free graphs has bounded clique-width only if is an induced subgraph of . In this section we survey the effect on boundedness of clique-width of restricting the class of -free graphs to just those graphs that belong to some hereditary graph class . Initially we do not want to make the hereditary graph class , in which we look for these -free graphs, too narrow. However, if we let be too large, the classification might remain the same as the one for general -free graphs in Theorem 4. This is the case if we let be the class of perfect graphs, or even the class of weakly chordal graphs, which form a proper subclass of perfect graphs by Observation 2.
[[BDHP17]] Let be a graph. The class of -free weakly chordal graphs has bounded clique-width if and only if is an induced subgraph of .
If we restrict further, then there are several potential classes of graphs to consider, such as chordal graphs, permutation graphs and distance-hereditary graphs (see also Figure 5). However, distance-hereditary graphs are known to have clique-width at most [GR00] (and hence their clique-width can be computed in polynomial time using the algorithm of [CHLRR12]). On the other hand, the classes of chordal graphs and permutation graphs have unbounded clique-width. This follows from combining Observation 2 with one of the following three theorems.
[[GR00]] The class of unit interval graphs has unbounded clique-width.
[[KLM14]] The class of split permutation graphs has unbounded clique-width.
[[BL03]] The class of bipartite permutation graphs has unbounded clique-width.
The case when is the class of chordal graphs has received particular attention, as we now discuss. Brandstädt, Engelfriet, Le and Lozin [BELL06] proved that the class of -free chordal graphs has unbounded clique-width. However, there are many graphs besides for which the class of -free chordal graphs has bounded clique-width. A result of [CR05] implies that -free chordal graphs have bounded clique-width for every integer . Brandstädt, Le and Mosca [BLM04] showed that -free chordal graphs have clique-width at most and that -free chordal graphs are distance-hereditary graphs and thus have clique-width at most . Brandstädt, Dabrowski, Huang and Paulusma [BDHP17] proved that bull-free chordal graphs have clique-width at most , improving a known bound of [Le03]. The same authors also proved that -free chordal graphs have clique-width at most , and that the classes of -free chordal graphs, -free chordal graphs and -free chordal graphs each have bounded clique-width.
Combining all the above results [BDHP17, BELL06, BLM04, CR05, GR00, MR99] leads to the following summary for -free chordal graphs; see Figure 7 for definitions of the graphs and and Figure 8 for pictures of all (maximal) graphs for which the class of -free chordal graphs is known to have bounded clique-width.
[[BDHP17]] Let be a graph with . The class of -free chordal graphs has bounded clique-width if and only if:
As can be seen from its statement, Theorem 4.1 leaves only two cases open, namely and ; see also [BDHP17].
Determine whether the class of -free chordal graphs has bounded or unbounded clique-width when or .
Recall that split graphs are chordal by Observation 2 and have been shown to have unbounded clique-width [MR99] (this also follows from Theorem 4.1). We now let be the class of split graphs, that is, we consider classes of -free split graphs, and find graphs for which the class of -free split graphs has bounded clique-width. We first note that as the class of split graphs is the class of -free graphs [FH77], the complement of a split graph is also a split graph by Observation 2. By Fact 2 this implies the following observation, which we discuss in more depth in Section 4.4.
For a graph , the class of -free split graphs has bounded clique-width if and only the class of -free split graphs has bounded clique-width.
Brandstädt, Dabrowski, Huang and Paulusma considered -free split graphs in [BDHP16]. They considered the two cases and that are open for -free chordal graphs (Open Problem 4.1) and proved that the classes of -free split graphs and -free split graphs have bounded clique-width. They showed the same result for -free split graphs, -free split graphs, -free split graphs and -free split graphs; see Figure 9 for a description of each of these graphs. They also proved that for every integer , the clique-width of -free split graphs is at most . Moreover, they showed the following: if is a graph with at least one edge and at least one non-edge that is not an induced subgraph of a graph in (see Figure 10), then the class of -free split graphs has unbounded clique-width. Note that both and have seven vertices. The -vertex induced subgraphs of are: and . The -vertex induced subgraphs of are: and . The above results lead to the following theorem.
[[BDHP16]] Let be a graph not in . The class of -free split graphs has bounded clique-width if and only if:
for some , or
is an induced subgraph of a graph in .
Determine whether the class of -free split graphs has bounded or unbounded clique-width when or .