Graphs of bounded cliquewidth are polynomially χ-bounded

10/01/2019 ∙ by Marthe Bonamy, et al. ∙ 0

We prove that if C is a hereditary class of graphs that is polynomially χ-bounded, then the class of graphs that admit decompositions into pieces belonging to C along cuts of bounded rank is also polynomially χ-bounded. In particular, this implies that for every positive integer k, the class of graphs of cliquewidth at most k is polynomially χ-bounded.

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1 Introduction

-boundedness.

A class of graphs is -bounded if there exists a function , called the -bounding function, such that for every graph . Here, is the chromatic number of — the least number of colors needed for a proper coloring of — and is the clique number of — the size of a largest set of pairwise adjacent vertices in .

The notion of -boundedness was proposed by Gyárfás in [Gyá87] and is a natural generalization of the concept of perfect graphs. Since then, many classes of not necessarily perfect graphs have been shown to be -bounded: examples include circular arc graphs [Gyá87], intersection graphs of axis-parallel boxes in -dimensional space [Gyá87], circle graphs [Gyá85, Gyá86]

, odd-hole-free graphs 

[SS16], long-hole-free graphs [CSS17], and graphs excluding an induced subdivision of a fixed tree [Sco97]. Many of those classes have natural interpretations as intersection graphs of geometric objects, but -boundedness is not an ubiquitous phenomenon in the geometric setting; for instance, the class of intersection graphs of segments in the plane is not -bounded [PKK14]. We refer the reader to the recent survey of Scott and Seymour [SS18] for a broader introduction to the topic.

While in the topic of -boundedness many basic questions still remain open, even less is known about the optimum asymptotics of -bounding functions. Here, we say that a class is polynomially -bounded if the -bounding function can be chosen to be a polynomial; we can define e.g. linear or quadratic -boundedness in the same way. Curiously, we do not know any hereditary graph class that would be -bounded, but (provably) not polynomially -bounded [Esp17]. For several concrete hereditary graph classes one can establish polynomial -boundedness, see the recent survey of Schiermeyer and Randerath [SR19]. However, in many important cases, including the advances on variants of the Gyárfás-Sumner conjecture and other conjectures of Gyárfás [CSS17, Sco97, SS16], the known proofs lead only to exponential upper bounds. Also, while the -boundedness of circle graphs was known since 80s [Gyá85, Gyá86], their polynomial -boundedness was established only very recently [DM19].

Cliquewidth.

The primary object of interest in this work is the class of graphs of cliquewidth at most , for any fixed . The cliquewidth of a graph is a parameter that measures the complexity of in terms of the number of labels needed to construct by means of certain algebraic operations. Roughly equivalently (up to a multiplicative factor of ), one can imagine that a graph of cliquewidth at most can be hierarchically decomposed into smaller and smaller pieces, up to single vertices, so that in each step of the decomposition we partition the current vertex set into two subsets so that vertices on each side of the partition have only at most different neighborhoods on the other side. This view is closely related to the notion of rankwidth of , where we measure the complexity of the partition as above not in terms of its diversity — the number of equivalence classes of the relation of having the same neighborhood on the other side — but in terms of its cutrank — the rank over of the adjacency matrix between the sides. Indeed, it is known that the cliquewidth of a graph of rankwidth is between and  [OS06]. Hence, a graph class has bounded rankwidth (i.e., the supremum of the rankwidth of its members is finite) if and only if it has bounded cliquewidth.

Dvořák and Král’ proved in [DK12] that for every , the class of graphs of cliquewidth at most is -bounded111Note that Dvořák and Král’ state their results in the setting of rankwidth instead of cliquewidth, but, as explained above, the boundedness of these two parameters is equivalent.. While the obtained -bounding function is not stated explicitly in [DK12], a careful examination of the proof shows that it is exponential in , even for constant .

In fact, Dvořák and Král’ in [DK12] prove a stronger claim, which can be informally stated as follows: if a graph class is -bounded, then for every , the class of graphs that admit decompositions into pieces belonging to along cuts of rank at most is also -bounded. While the result for graphs of bounded cliquewidth follows by applying this statement for a trivial class , one can also obtain other corollaries. For instance, by composing their main result with a decomposition theorem of Geelen for graphs excluding the wheel as a vertex minor [Gee95], Dvořák and Král’ conclude that this graph class is -bounded. Here, the wheel consists of a cycle on vertices plus one additional vertex connected to all of them.

Our results.

In this work we prove that every graph class of bounded cliquewidth is not only -bounded, but in fact polynomially -bounded.

Theorem 1.1.

For every , the class of graphs of cliquewidth at most is polynomially -bounded.

Let us note that in Theorem 1.1, the degree of the -bounding polynomial for graphs of cliquewidth grows with . As we explain in Section 5.1 (see Lemma 5.2), this is unavoidable.

We also remark that independently of us, Nešetřil et al. proved that for every fixed , graphs of linear cliquewidth at most are linearly -bounded [NORS19]. In their work, the result comes as a by-product in the proof of a statement about logical interpretability of graphs of bounded linear cliquewidth with no large half-graphs in graphs of bounded pathwidth. They provide two proofs of their main claims, one using the same tool as we do — Simon’s factorization — and second more direct and involved, but yielding better asymptotic bounds. In fact, a close inspection shows that their first proof restricted to arguing only linear -boundedness is essentially equivalent to our proof restricted to classes of bounded linear cliquewidth, with the exception that a suitable amortization argument (disguised as applying the fact that cographs are perfect) is used to argue that the -bounding function is linear, instead of just polynomial.

In fact, our work applies not only to classes of bounded cliquewidth. We prove a more general claim analogous to that of Dvořák and Král’: if is a polynomially -bounded graph class and , then the class of graphs that can be decomposed into pieces from along cuts of diversity at most is also polynomially -bounded. A formal statement of this result requires some technical definitions and will be presented in Section 2, see Theorem 2.4 there.

It seems that our main result may be applicable for establishing polynomial -boundedness of classes defined by forbidding fixed vertex-minors, similarly to the work of Dvořák and Král’ [DK12]; we discuss these connections in Section 5.2. In particular, we generalize two related statements that were used in this context: the result of Chudnovsky et al. [CPST13] that the closure of a polynomially -bounded class under the substitution operation is also polynomially -bounded, and the recent result of Kim et al. [KKOS19] that the same holds also for the operation of taking -joins. Indeed, these two cases follow from taking and in our main theorem, respectively. In fact, in our proof we rely on the result of Chudnovsky et al. [CPST13].

Our techniques.

The first ingredient of the proof of our main result is the deterministic variant of Simon’s Factorization Forest Theorem due to Colcombet [Col07], which provides a factorization theorem for trees labelled with elements of a finite semigroup. The idea is that we can view a decomposition of a graph using cuts of diversity at most as a tree labelled with elements of the semigroup of relabelings (functions) on a set of labels. Thus, by applying Colcombet’s result we can hierarchically factorize every such decomposition so that the following properties hold:

  • the factorization has “depth” ; and

  • in every step we partition the current decomposition into factors (which will be factorized further in the following steps) so that the overall structure of factors is a tree that either has depth at most two, or satisfies a certain Ramsey-type condition.

Thus, we can reduce the original statement of the main result to applications of a weaker statement, where the provided decomposition of the graph either has depth at most two or enjoys strong Ramsey properties. Decompositions of the former type are called shallow, and decompositions of the latter type are called splendid.

While the case of decompositions of depth two follows from a straightforward product argument, the case of splendid decompositions requires a non-trivial reasoning. For this, we use the aforementioned result of Chudnovsky et al. [CPST13] about closures of polynomially -bounded classes under the substitution operation. The splendidness of the decomposition implies that if we partition edges of the graph into those “introduced” at odd and even levels, in both cases we observe a subgraph that can be obtained from graphs from the base class by a repeated application of substitutions. Hence, we can apply the result of Chudnovsky et al. [CPST13] to both these subgraphs and conclude by taking the product of the obtained colorings.

2 Preliminaries

For a positive integer , we write .

Graph terminology.

In this work we consider only finite, undirected graphs. For a graph , the vertex and edge sets of are denoted by and , respectively. The neighborhood of a vertex consists of all vertices adjacent to and is denoted by . For a graph , we write for the size of the largest clique in , i.e., a set of pairwise adjacent vertices. A coloring of is any function that maps vertices of to some set of colors, and it is proper if for every edge of , the endpoints of receive different colors in the coloring. For a graph , the chromatic number of , denoted , is the least number of colors needed for a proper coloring of .

A graph class is just a set of graphs, usually infinite. A class of graphs is hereditary if it is closed under taking induced subgraphs, that is, if then every graph obtained from by deleting vertices also belongs to . A graph class is -bounded if there exists a function such that for every . If the function can be chosen to be a polynomial, then we say that is polynomially -bounded.

A rooted tree is a connected graph without cycles with one vertex designated to be the root. As all decompositions in this work will have a form of rooted trees, for distinguishment we usually use the term node for a vertex of a tree. If is a rooted tree and are its nodes, then we say that is an ancestor of , equivalently that is a descendant of , if lies on the unique path in from to the root. Note that every node is considered its own ancestor and descendant.

Cliquewidth.

A -labelled graph is a graph together with a labelling . On -labelled graphs we define the following operations:

  • for , , operation adds all possible edges with one endpoint of label  and second of label ;

  • for , , operation changes the label of each vertex labelled to ;

  • operation outputs the disjoint union of two -labelled graphs.

Note that and are unary operations, that is, they are applied to a single -labelled graph, while is a binary operation. The cliquewidth of an (unlabelled) graph is the least integer such that some -labelling of can be constructed using the operations described above from single-vertex -labelled graphs.

Note that the construction of a graph of cliquewidth , as described above, naturally gives rise to a term over an algebra of operations , , and whose leaves are single-vertex -labelled graphs. This term is called a -expression that constructs .

Decompositions.

As outlined in Section 1, we will actually prove a result stronger than claimed in Theorem 1.1: whenever is a class of graphs that is polynomially -bounded, the class of graphs that admit decompositions using cuts of bounded diversity into pieces that belong to is also polynomially -bounded. To state this result formally, we need to introduce the notion of a decomposition that will be used. It is actually a rooted variant of -bounded decompositions used by Dvořák and Král’ [DK12], hence we use a similar terminology, but adjusted to the rooted setting.

Definition 2.1.

A decomposition of a graph is a rooted tree together with a function that maps vertices of to nodes of .

Note in the above definition, there are no requirements about surjectivity or injectivity of the mapping . In particular, many vertices of may be mapped to the same node of . Also, there are no restrictions on the number of children of any node in .

Whenever a decomposition of a graph is clear from the context, we use the following notation. For two nodes of , we write if is an ancestor of in ; recall that we consider every node to be its own ancestor as well. For an edge of , we write

For a node of , we write

If the decomposition for which the above objects are defined is not clear from the context, we write it as the second argument in the brackets, i.e. we write .

Let us remark that we believe that the graph , as defined above, may be the right analogue of the torso of a node of from the context of classic tree decompositions. Note that for every child of , the vertices of form an independent set in .

Definition 2.2.

For a class of graphs , a decomposition of a graph is -governed if for every node of .

For a node of , we introduce the following equivalence relation on :

In other words, are considered equivalent if they have exactly the same neighborhood outside of . Note that if is a descendant of , then entails .

Definition 2.3.

Let be a decomposition of a graph . The diversity of a node of is the number of equivalence classes of . The diversity of the decomposition is the largest diversity among the nodes of .

Statement of the results.

With all the terminology prepared, we can formally state the main result of this paper.

Theorem 2.4.

Let be a hereditary graph class that is polynomially -bounded. Then for every fixed , the class of graphs that admit -governed decompositions of diversity at most is also polynomially -bounded.

Now, Theorem 1.1 is directly implied by Theorem 2.4 and the next lemma, which follows almost immediately from the definition of cliquewidth.

Lemma 2.5.

For every , every graph of cliquewidth at most admits a decomposition of diversity at most governed by the class of bipartite graphs.

Proof.

Let be a graph of cliquewidth at most and let be a -expression constructing (some -labelling of) . Let be the rooted tree of the -expression , i.e., the tree of the subterms of with the descendant relation defined by the subterm relation. Note that the vertices of are in one-to-one correspondence with the leaves of where they are introduced, hence let map every vertex of to the corresponding leaf of . Then is a decomposition of .

Observe that for every node of , say corresponding to a subterm of , the set of vertices introduced in the descendants of is exactly . By the definition of a -expression, all vertices assigned the same label in the graph constructed by have the same neighborhood outside of in , hence has at most equivalence classes. It follows that has diversity at most .

To see that is governed by the class of bipartite graphs, observe that every internal node has at most two children and , hence is bipartite. On the other hand, if is a leaf then has one vertex, hence it is bipartite as well. ∎

As the proof of Theorem 2.4 spans the whole remainder of the paper, from now on we fix the integer and the hereditary graph class . Also, we write for the latter class considered in Theorem 2.4, i.e. the class of graphs that admit -governed decompositions of diversity at most .

3 Stratifying the class

The first step in the proof of Theorem 2.4 is to stratify the inclusion . That is, we introduce a sequence of classes , for some depending only on , with the following property: for every , every graph admits a -governed decomposition of diversity at most

that is moreover somehow well-behaved, to be defined in a moment. Then Theorem 

2.4 can be proved by an induction over the sequence , where every step of the induction boils down to applying Theorem 2.4 under the additional supposition of well-behavedness of the decomposition. Thus, we reduce Theorem 2.4 to a weaker statement where we have an additional assumption about the decomposition.

To formally explain “well-behavedness” we need some additional terminology. We let be the semigroup of all functions from to , with composition being the semigroup action. That is, for we write for the function that maps every to .

Definition 3.1.

Suppose is a decomposition of a graph with diversity at most . A tagging of is a family such that every is a mapping from to with the following property: whenever for some , then .

With a tagging of a decomposition of we associate a labelling defined as follows: for every edge of , say where is the parent of , we set to be the function that maps every to the unique such that for every satisfying .

Note that every decomposition of diversity at most has a tagging: any tagging can be obtained by enumerating the classes of with numbers from to , in any way. We will, however, consider decompositions that admit taggings which give rise to somewhat restricted labellings, as explained next.

Definition 3.2.

Let be a semigroup. A set of elements is forward Ramsey if for all we have . In particular, each is an idempotent in , that is, .

Definition 3.3.

A decomposition of a graph of diversity at most is called splendid if there exists a tagging of such that the set is forward Ramsey in .

The next lemma formalizes the idea of stratification of the inclusion and is the main result of this section.

Lemma 3.4.

There exists and a sequence of hereditary graph classes

such that for every , every graph admits a -governed decomposition of diversity at most that is either splendid or shallow.

Using Lemma 3.4 and straightforward induction, the proof of Theorem 2.4 boils down to proving the statement under the additional assumption that the decomposition witnessing is either splendid or shallow. We treat this case in Section 4, while for the rest of this section we concentrate on proving Lemma 3.4.


Our main tool for the proof of Lemma 3.4 is the deterministic variant of Simon’s Factorization Forest Theorem, due to Colcombet [Col07]. This result originates in the algebraic theory of formal languages, so to state it we need several auxiliary definitions.

Deterministic Simon’s factorization.

In the following, a word over a semigroup is a (possibly empty) sequence of elements of . The set of all words over is denoted by , whereas denotes the set of all non-empty words over . For a word of length and positions , we write for the subword of starting with the symbol at position and ending with the symbol at position , where positions are numbered from to .

The concatenation of two words and will be denoted by . Note that endowed with concatenation is an (infinite) semigroup, with a natural homomorphism defined as follows: for , computes the product in of the symbols appearing in from left to right.

Suppose is a word over , say of length . A split of of height is a mapping . Two positions are -equivalent, denoted , if

A split of is called forward Ramsey if the following condition is satisfied: for all quadruples of positions that are pairwise -equivalent and satisfy and , we have

With this terminology introduced, we can state the result of Colcombet.

Theorem 3.5 (Theorem 1 of [Col07]).

For every finite semigroup there exists a mapping satisfying the following property. For a word , say of length , define a split of as follows: for , set . Then for every , the split is forward Ramsey.

We remark that in [Col07] a stronger conclusion is provided, namely that the value of can be computed by running a deterministic automaton with at most states over . We will not use this property here.

Factorizing trees.

As observed in [Col07], Theorem 3.5 is well-suited for the treatment of trees. Essentially, it proves that for any rooted tree whose edges are labelled with elements of a finite semigroup, we can find a bounded-depth hierarchy of forward Ramsey factorizations. This idea is formalized next.

Fix a finite semigroup . An -labelled tree is a rooted tree together with a mapping , called further the labelling. Similarly as before, an -labelled tree is splendid if the set of labels is forward Ramsey in .

A factorization of a rooted tree is a partition of such that every part of induces a connected subtree of . These subtrees are called the factors of and for a factor of , we let be the top-most vertex of . Thus, is the set of all top-most vertices of factors of . If is -labelled, then we consider the factors of to be -labelled as well by restricting the labelling of .

For a factorization of a rooted tree , we define the quotient tree as follows: the node set of is and the ancestor/descendant relation in is the restriction of this relation from to . Note that this is equivalent to contracting every factor of into a single node named , and setting the ancestor/descendant relation between contracted factors in the obvious manner.

If is a factorization of and is -labelled, say with a labelling , then we can define a labelling of as follows. Take any edge of , say where is the parent of in . Then is an ancestor of in and let be the path in from to . If are the consecutive edges of on , then we let

Now, from Theorem 3.5 we can derive the following Simon-like result for trees.

Lemma 3.6.

For every finite semigroup there exists a sequence of classes of -labelled trees

satisfying the following conditions:

  • consists only of one tree with one node, while is the class of all -labelled trees; and

  • for every , every tree has a factorization such that all factors of belong to and is either splendid or shallow.

Proof.

We will use the following notation. For an -labelled tree and node of , we write for the -labelled tree induced in by the descendants of (including itself); here is the restriction of to edges between those descendants. More generally, if is a subtree of , then is the restriction of to the edges between the nodes of . Also, we write for the (oriented) path in that starts at the root of and ends at . Then is the word consisting of elements of assigned by to consecutive edges of . Note that if is the root of , then is the empty word.

For an -labelled tree and positive integer , any function will be called a split of of height . Such a split is forward Ramsey if the following holds: for every node of , the split restricted to the nodes on induces a forward Ramsey split of .

We define the level of as follows:

From Theorem 3.5 we easily deduce that the function has bounded range.

Claim 1.

For every -labelled tree we have .

Proof.

It suffices to construct a split of of height at most . Let be the mapping provided by Theorem 3.5 for the semigroup . Then let us define a split as follows: for every we set . Then Theorem 3.5 directly implies that is a forward Ramsey split of .    

On the other hand, we inductively define the notion of the complexity of an -labelled tree as follows:

  • If has one node then .

  • Otherwise, is the least positive integer such that admits a factorization where every factor of has complexity smaller than and is either splendid or shallow.

It is not hard to see that every -labelled tree has finite complexity; say, bounded by its depth. However, this will follow from the following statement, which is the core argument of the proof.

For every -labelled tree we have . ()

Note that establishing will conclude the proof, because we will define classes as follows: comprises of all -labelled trees of complexity at most . Then Claim 1 and () ensure us that contains all -labelled trees, while from the definition of the complexity we infer that consists only of the one-node tree and that the second condition from the lemma statement holds.

Therefore, from now on we focus on proving (). For this, we proceed by induction on the level of .

Base case.

Suppose has level , which means that it admits a forward Ramsey split that such that for each node of . The following generic claim will be useful.

Claim 2.

Suppose are forward Ramsey. If , then is forward Ramsey as well.

Proof.

We need to verify that for any , we have . If or then this follows from the assumption that and are forward Ramsey. Hence, by symmetry suppose that and . Take any . Since and is forward Ramsey, we have

Therefore,

as required.    

Let be the root of . For every child of , let be the set of all elements of that are assigned by to the edges of the subtree of induced by the descendants of .

Claim 3.

For every child of , the set is forward Ramsey.

Proof.

Since the split is forward Ramsey, for every descendant of the set of elements appearing in is forward Ramsey. All these sets for different descendants contain the element . Hence, by repeated application of Claim 2 we conclude that is forward Ramsey, hence is forward Ramsey as well.    

We observe that for every child of , the tree has complexity at most . Indeed, we can take its factorization into single-node factors, which have complexity , and then is splendid due to Claim 3. Then has complexity at most , because we can take its factorization that puts the root as a one-node factor and each subtree , for ranging over children of , as a separate factor. Indeed, then has depth at most so is shallow. As , this concludes the base case of the induction.

Induction step.

Suppose that is an -labelled tree of level . Let be the forward Ramsey split of that witnesses . Further, let .

Define an equivalence relation on as follows: are -equivalent if either both do not have an ancestor belonging to , or they have the same least ancestor belonging to . It is easy to see that the equivalence classes of induce connected subtrees of , hence the partitioning of according to the equivalence classes of is a factorization of . For each , we let be the factorization of consisting only of those factors of that contain only descendants of .

We first observe that a reasoning similar to that of Claim 3 shows that for “deep” elements , is already a good factorization of .

Claim 4.

Suppose a node has an ancestor that belongs to and is different from . Then is splendid.

Proof.

Let be the least ancestor of such that and . Consider any descendant of that belongs to and let be the path in from to ; then is the first edge of . Since is forward Ramsey, , and all values of are not larger than , it follows that the set of elements of assigned by to the edges of is forward Ramsey; call this set . Observe that sets pairwise share the element , hence by repeated application of Claim 2 we conclude that is forward Ramsey. Since this set contains , we conclude that is splendid.    

Claim 5.

For every node that has an ancestor belonging to but different from , we have .

Proof.

By Claim 4, admits factorization such that is splendid. Therefore, it suffices to prove that each factor of has complexity at most .

Observe that the only node of mapped to by is the root of . Therefore, consider the factorization of that puts the root of into a one-node factor and every subtree of rooted at a child of the root into a separate factor. Observe that for every such subtree , restricting to yields a forward Ramsey split of of height at most , witnessing that . By induction hypothesis we have . As is shallow and the root factor of has complexity , we conclude that has complexity at most .    

Let now be the connected subtree of induced by those nodes of that have at most one ancestor belonging to (including themselves). Note that contains the root of .

Claim 6.

.

Proof.

First observe that for every , in the tree — the subtree of induced by all descendants of that belong to — the only node mapped to by is the root. Therefore, the same argument as in the proof of Claim 5 shows that .

Now if the root of belongs to , then , hence , as claimed. Otherwise, let be the connected subtree of induced by all those nodes of that have no ancestor in (including themselves); then contains . Note that restricted to is a forward Ramsey split of of height at most , hence by the induction hypothesis we have . Now consider a factorization of consisting of factor and factors for all . Since all these factors have complexity at most and has depth at most , we conclude that , as claimed.    

Now, construct a factorization of as follows: the factors of consists of and for each such that has exactly one ancestor that belongs to and is different from . By Claims 5 and 6, each of the factors of has complexity at most . Furthermore, has depth at most . This implies that has complexity at most , which concludes the proof of () and of Lemma 3.6. ∎

With Lemma 3.6 established, the conclusion of the proof of Lemma 3.4 boils down to an easy verification.

Proof of Lemma 3.4.

Apply Lemma 3.6 to the semigroup , yielding a suitable sequence of classes of -labelled trees . We set

and define graph classes as follows: a graph belongs to if and only if there exists a -governed decomposition of of diversity at most and its tagging such that , treated as an -labelled tree, belongs to . We now verify that classes defined in this way satisfy all the required properties.

First, observe that graphs from are exactly those that admit -governed decompositions with a one-node decomposition tree. Thus . On the other hand, from Lemma 3.6 we conclude that . Also, since is hereditary, so are all the classes .

Now, fix any and consider any graph . Let be a decomposition and be its tagging that witness the membership . Then . By Lemma 3.6 we infer that there exists a factorization of such that every factor of belongs and is either splendid or shallow.

Define a decomposition of as follows: for every , we set to be the top vertex of the unique factor of that contains . Let be the restriction of the tagging to the nodes of , which is then a tagging of . The following claim then follows from definitions in a straightforward manner; here, is the labelling induced by tagging in .

Claim 1.

It holds that .

It now suffices to check that the decomposition posesses all the properties asserted in the lemma statement. Note that for every node of we have , hence we use the shorthand for both.

Claim 2.

has diversity at most .

Proof.

Since for every node we have , also the relation over computed in is the same as computed in . As has diversity at most , this relation has always at most equivalence classes, which implies that also has diversity at most .    

Claim 3.

is -governed.

Proof.

Consider any node of . Then for some factor of . Let be defined as follows: for we put

  • , if belongs to ; and

  • to be the least ancestor of that belongs to , otherwise.

Let that is, the vertex set of is and the edge set of consist of all edges of such that the least common ancestor of and in belongs to . Then is a decomposition of . A similar argument as in Claim 2 shows that the diversity of is at most . Also, as a decomposition of is -governed, because

Moreover, if is the restriction of the tagging to the nodes of and is the restriction of the labelling to the edges of , then it is easy to see that . Since , we infer that together with tagging witnesses that . As was taken arbitrarily, we conclude that is -governed.    

Claim 4.

is either splendid or shallow.

Proof.

Recall that , as an -labelled tree, is either splendid or shallow. In the latter case we can immediately conclude, because then the decomposition has depth at most . In the former case, by Claim 1 we infer that is splendid. This means that the tagging witnesses that is splendid.    

The above three claims verify that has the required properties and we are done. ∎

4 Treating shallow and splendid decompositions

4.1 Shallow decompositions

For shallow decompositions we may use a direct product argument.

Lemma 4.1.

Let be a class of graphs that is -bounded by a non-decreasing function . Then the class of graphs that admit -governed shallow decompositions is -bounded by the function .

Proof.

Let be a graph that admits a decomposition that is -governed and shallow. Let be the root of ; then . By assumption and for every child of . Hence, we can find a proper coloring of with at most colors, and, for every child of , a proper coloring of with at most colors. Then define a coloring of as follows: for any , we let

Clearly, uses at most colors, hence it suffices to verify that is a proper coloring of .

Consider any edge of . If , then also and the edge is present in the graph . As is a proper coloring of , the second coordinates of and differ. Otherwise, if , then , because has depth at most . Then the edge is present in the graph and, since is a proper coloring of , we conclude that the first coordinates of and differ. In both cases we have , hence is a proper coloring of . ∎

4.2 Splendid decompositions

For splendid decompositions, the main tool will be a result of Chudnovsky et al. [CPST13], which treats of the closure of classes of graphs under the operation of substitution, defined as follows. Suppose that is a graph and is a mapping that associates with each vertex of some graph . Then we define the graph as follows:

  • The vertex set of consists of pairs of the form , where is a vertex of and is a vertex of .

  • Two vertices and are adjacent in if either and is an edge in , or and is an edge in .

Informally speaking, is obtained from by replacing every vertex with the graph , and putting a complete join between graphs and whenever was an edge of .

For a graph class , let be the closure of under the substitution operation. That is, is the smallest class that contains and whenever and the image of is contained in , then as well. The following claim links the operation of substitution with our terminology and is easy to verify; we leave it without a proof, as we will not use it later on.

Lemma 4.2.

For every hereditary graph class closed under substituting vertices with edgeless graphs, the class is exactly the class of all graphs that admit a -governed decomposition of diversity .

Chudnovsky et al. [CPST13] proved the following.

Theorem 4.3 (Theorem 2.3 of [Cpst13]).

If a hereditary graph class is polynomially -bounded, then so is .

Thus, by Lemma 4.2, Chudnovsky et al. in fact essentially proved Theorem 2.4 for . We will now use this result to prove Theorem 2.4 in the case when the provided decomposition is splendid.

Lemma 4.4.

Let be a hereditary class of graphs that is polynomially -bounded. Then the class of graphs that admit splendid -governed decompositions of diversity at most is also polynomially -bounded.

Proof.

First, we need to understand forward Ramsey sets in the semigroup .

Claim 1.

Suppose is forward Ramsey. Then there exists an equivalence relation on such for every and all satisfying , we have and .

Proof.

Fix any and define the equivalence relation as follows: for , we have if and only if . To verify that defined in this way has the required property, consider any and any such that , that is, . Since due to being forward Ramsey, we have

Similarly we have , hence

implying that .    

We proceed to the main proof. We may assume that contains all edgeless graphs, because adding all such graphs to does not spoil the assumption that is hereditary and polynomially -bounded.

Consider any graph that admits a splendid -governed decomposition of diversity at most . Let be a tagging of that witnesses that is splendid, that is, the set is forward Ramsey. Let be the equivalence relation on provided by Claim 1 for this set.

Claim 2.

For every and satisfying and , we have .

Proof.

Note that the nodes satisfying are exactly the ancestors of . As these ancestors induce a path in , by the transitivity of it suffices to prove the claim in the case when and are adjacent in , say is the parent of .

Let