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Polynomial Kernels for Hitting Forbidden Minors under Structural Parameterizations

We investigate polynomial-time preprocessing for the problem of hitting forbidden minors in a graph, using the framework of kernelization. For a fixed finite set of connected graphs F, the F-Deletion problem is the following: given a graph G and integer k, is it possible to delete k vertices from G to ensure the resulting graph does not contain any graph from F as a minor? Earlier work by Fomin, Lokshtanov, Misra, and Saurabh [FOCS'12] showed that when F contains a planar graph, an instance (G,k) can be reduced in polynomial time to an equivalent one of size k^O(1). In this work we focus on structural measures of the complexity of an instance, with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large. Motivated by several impossibility results, we parameterize the F-Deletion problem by the size of a vertex modulator whose removal results in a graph of constant treedepth η. We prove that for each set F of connected graphs and constant η, the F-Deletion problem parameterized by the size of a treedepth-η modulator has a polynomial kernel. Our kernelization is fully explicit and does not depend on protrusion reduction or well-quasi-ordering, which are sources of algorithmic non-constructivity in earlier works on F-Deletion. Our main technical contribution is to analyze how models of a forbidden minor in a graph G with modulator X, interact with the various connected components of G-X. By bounding the number of different types of behavior that can occur by a polynomial in |X|, we obtain a polynomial kernel using a recursive preprocessing strategy. Our results extend earlier work for specific instances of F-Deletion such as Vertex Cover and Feedback Vertex Set. It also generalizes earlier preprocessing results for F-Deletion parameterized by a vertex cover, which is a treedepth-one modulator.

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1 Introduction

How, and under which circumstances, can a polynomial-time algorithm prune the easy parts of an NP-hard problem input, without changing its answer? This question can rigorously be answered using the notion of kernelization [2, 24, 30] which originated in parameterized complexity theory [9, 13] where it can be naturally framed. After choosing a complexity parameter for the NP-hard problem of interest, which associates to every input  an integer  that expresses its difficulty under the chosen type of measurement, the theory postulates that a good preprocessing algorithm can be captured by the notion of a polynomial kernelization: a polynomial-time algorithm that, given a parameterized instance , outputs an instance  with the same answer whose size is bounded polynomially in . Not all parameterized problems admit polynomial kernelizations, and one can find meaningful ways to preprocess an NP-hard problem by studying those parameterizations for which it does. The study of kernelization has blossomed over the last decade, resulting in a myriad of interesting techniques for obtaining polynomial kernelizations [4, 16, 25, 32, 35], as well as frameworks for proving the non-existence of polynomial kernelizations under complexity-theoretic assumptions [2, 3, 12, 14, 21].

Originally, the study of kernelization focused on the natural parameterizations of (the decision variants of) search problems, where the complexity parameter  measures the size of the solution. A classic example [8, 36] is that an instance  of the -Vertex Cover problem, which asks whether an undirected graph  has a vertex cover of size , can efficiently be reduced to an equivalent instance with at most  vertices. This guarantees that efficient pruning can be done on large inputs that have small vertex covers. However, such guarantees are meaningless when the smallest vertex cover contains more than half the vertices. By choosing a parameter that measures the structure of the input graph, rather than the size of the desired solution, one can hope to develop provably good preprocessing procedures even for inputs whose solutions are large. An early example of this approach was given by Jansen and Bodlaender [27], who showed that an instance of the Vertex Cover problem can efficiently be reduced to size , where  is the size of a smallest feedback vertex set in : Vertex Cover parameterized by the size of a feedback vertex set has a cubic-vertex kernel. The result effectively conveys that large instances of Vertex Cover that are  vertex-deletions away from being acyclic, can be shrunk to size  in polynomial time.

Problem statement

To understand the power of polynomial-time preprocessing algorithms over inputs to NP-hard problems that exhibit some structural regularities, but whose solutions are generally large, we set out to answer the following question:

For which structural parameterizations of NP-hard graph problems is it possible to obtain polynomial kernelizations?

Our goal is to answer this question for a rich class of problems, in terms of a rich class of structural parameterizations. Existing lower bounds show that, in general graphs, it is unlikely that a logical characterization exists of the problems admitting polynomial kernelizations for structural parameterizations (cf. [17, §1]), even though meta-theorems in terms of logical definability or finite integer index are possible when dealing with inputs from sparse graph families [4, 22]. We therefore target the class of -Minor-Free Deletion problems, henceforth abbreviated as -Deletion problems, to capture a wide class of NP-hard graph problems. Such a problem is instantiated by specifying a finite set  of forbidden minors. An input then consists of a graph  and integer , and asks whether it is possible to find a set  of size  such that  contains no graph from  as a minor. This is a rich class of problems: by choosing  we obtain Vertex Cover, for  we have Feedback Vertex Set, and for  we obtain the problem of making a graph planar by vertex deletions. The kernelization complexity of the solution-size parameterization of -Deletion has been the subject of intensive research [18, 19, 23, 29, 41]. In this work we attempt to find the widest class of structural parameterizations for which -Deletion admits polynomial kernels, continuing a long line of investigation into structural parameterizations for Vertex Cover [5, 20, 27, 31, 32, 34], Feedback Vertex Set [28, 33], and other -Deletion problems [17, 22].

When it comes to measuring graph complexity, a natural choice is to consider a width measure such as treewidth. Alas, it has long been known that even Vertex Cover, the simplest -Deletion problem, does not admit a polynomial kernelization when parameterized by the treewidth of the input graph.111Bodlaender et al. [3, Theorem 1] show a superpolynomial kernelization lower bound for Independent Set parameterized by treewidth. Since the parameter is not related to the solution size, this is equivalent to Vertex Cover parameterized by treewidth. The lower bound holds under the assumption that , which we implicitly assume when stating further lower bounds in this section. Generally speaking, graph problems do not admit polynomial kernels under parameterizations that attain the maximum, rather than the sum, of the values of the connected components. We therefore use the vertex-deletion distance to simple graph classes  as the parameter. The aforementioned result by Jansen and Bodlaender [27] shows that Vertex Cover has a polynomial kernelization when parameterized by the vertex-deletion distance to an acyclic graph, i.e., to a graph of treewidth one. Unfortunately this formulation leaves little room for generalizations: no polynomial kernelization is possible parameterized by the distance to a graph of treewidth two [11, Theorem 11], or even pathwidth two.222The lower bound is stated for distance to treewidth two, but the same proof works for pathwidth two. We therefore cannot use the deletion distance to constant treewidth (tw) or pathwidth (pw) as our graph parameter, and use the deletion distance to constant treedepth (td) instead. The parameter treedepth has recently attracted much interest [7, 15, 39], sometimes allowing better upper bounds than are possible in terms of treewidth [22, 38]. It plays an important role in the study of structural sparsity [37]. All graphs  satisfy , so graphs of constant treedepth are more restricted than those of constant treewidth. We therefore study the following problem for a fixed set  of connected graphs and constant .

-Deletion parameterized by treedepth- modulator Parameter: . Input: A graph , integer , and a modulator  such that . Question: Is there a set  of size  such that  is -minor-free?

The restriction that  contains only connected graphs is needed to ensure that a solution on a disconnected graph can be formed from solutions on its connected components, which we require in some of our proofs. This restriction was also considered in previous work [19] on kernelization, but can be avoided when targeting single-exponential FPT algorithms [29].

For technical reasons, we assume that a modulator  is given in the input. If no modulator is known, one can compute an approximate modulator and use it as . For example, Gajarský et al. [22, Lemma 4.2] showed that a modulator of size at most  times the optimum can be found in quadratic time. Our problem setting is related to that of Gajarský et al. [22]. They studied kernelization for a general class of graph problems that includes -Deletion, parameterized by a constant-treedepth modulator, but under the additional restriction that the input graph has bounded expansion or is nowhere dense. Under this severe restriction they obtained kernelizations of linear size for a wide range of problems. This prompted Somnath Sikdar during the 2013 Workshop on Kernelization [10] to ask which types of problems admit polynomial kernelizations in general graphs, when parameterized by a constant-treedepth modulator; we address this question in this work.

Our results

Our main result proves the existence of polynomial kernelizations for -Deletion parameterized by a modulator whose removal leaves a graph of constant treedepth. For every fixed finite set  of connected graphs and every constant , the -Deletion problem parameterized by a treedepth- modulator has a polynomial kernelization. This answers a question posed by Bougeret and Sau [5] (cf. [6]). They obtained polynomial kernels for Vertex Cover parameterized by a constant-treedepth modulator, and asked whether their result can be extended to the Feedback Vertex Set problem. As Feedback Vertex Set is an -Deletion problem for , Theorem 1 shows that this is indeed the case. Theorem 1 greatly generalizes an earlier result of Fomin, Jansen, and Pilipczuk [17, Corollary 1], who proved that -Deletion parameterized by a vertex cover has a polynomial kernel for every fixed ; note that a vertex cover is precisely a treedepth- modulator.

Our kernelization is fully explicit and does not depend on protrusion replacement techniques or well-quasi-ordering, which are sources of algorithmic non-constructivity in other works [18, 19] on kernelization for -Deletion. Moreover, our general theorem allows  to be any set of connected graphs, including nonplanar ones. In contrast, the kernelization for the solution-size parameterization by Fomin et al. [19] only applies when  contains at least one planar graph. Hence they only capture problems where, after removing a solution, the remaining graph has constant treewidth [40]. In our case, even though the parameter value is expressed in terms of a modulator to a graph of constant treedepth and therefore constant treewidth, the graphs that result after removing an optimal solution may have unbounded treewidth. This occurs, for example, when using  to capture the Vertex Planarization problem. (Whether the solution-size parameterization of Vertex Planarization has a polynomial kernel is a notorious open problem [19].)

The degree of the polynomial in the kernel size bound grows very quickly with . We prove that this is unavoidable, even for the simplest case of Vertex Cover.

For every , the Vertex Cover problem parameterized by the size of a given treedepth- modulator  does not admit a kernelization of bitsize  for any , unless .

Techniques

To obtain a polynomial kernel for an instance  of -Deletion, the main challenge is to understand how the connected components  of  interact through their connections to the modulator . Using the language of labeled minors, we analyze how minor models of a forbidden graph in  may intersect the various components of . Using these insights, we are able to characterize which components of  affect the structure of optimal solutions in an essential way. On a high level, the kernelization strategy is as follows. We use the fact that a single constant-treedepth component can be analyzed efficiently, to identify a subset  of  that contains  essential components under our characterization. We prove that the remaining ones can be safely removed, because their interaction with the rest of the instance can be ignored. Formally speaking, we show that any optimal solution on  can be lifted to a solution on  by including  additional vertices:  is a yes-instance if and only if  is. This effectively shows that there is an optimal solution  on  in which the non-essential components act in isolation:  does not delete more vertices from such a component, than would be deleted by a solution on the graph .

The overall kernelization follows straight-forwardly from this pruning of non-essential components by a recursive approach, similarly as in earlier work [5, 22]. The main challenge is therefore to understand which components are essential and which are not, and this is where our contribution lies. We present a stand-alone combinatorial lemma that captures our key insight in this direction. To state it, we introduce some terminology.

We work with a nonstandard notion of labeled graphs. For a finite set , an -labeled graph is a graph in which each vertex is assigned a (possibly empty) subset of  as its labelset; we stress that multiple vertices may carry the same label on their labelset. The minor relation on graphs extends to labeled graphs in a natural way: a labeled graph  is a minor of a labeled graph , if  can be obtained from  by repeatedly deleting an edge, deleting a vertex, deleting a label from the labelset of a vertex, or contracting an edge. When contracting an edge  into a single vertex , the labelset of  is formed as the union of the labelsets of  and .

For a collection  of vertex subsets  of an -labeled graph , and a set of -labeled graphs , we say that all  leave a -minor in , if for all  the graph  contains some graph  as a labeled minor. We say that a set  of -labeled graphs is -saturated for an integer , if for each subset  of size , the graph consisting of one vertex with labelset  belongs to . Our main lemma states that if all optimal solutions to -Deletion on  leave a -minor for some suitably saturated , then there is a small subset  for which the same holds.

[Main lemma] Let  be a finite set of (unlabeled) connected graphs, let  be a set of labels, let  be a -saturated set of connected -labeled graphs of at most  vertices each, and let  be an -labeled graph. If all optimal solutions to -Deletion on  leave a -minor, then there is a subset  whose size depends only on , such that all optimal solutions leave a -minor.

In several aspects, the statement in the lemma is best-possible. In particular, we will show in Section 3 that the dependence of the size of  on  rather than  is essential and that the precondition that  is -saturated cannot be avoided.

Lemma 1 is the cornerstone in our understanding of which components of  are essential. In our applications of the lemma, the graph  consists of a connected component of  whose labels encode the adjacency of those vertices to the modulator . The set  contains potential fragments of models of forbidden -minors, again labeled by adjacency to , which we may be interested in destroying in  so that connections through  cannot form -minors with fragments that remain in other components of . The lemma then essentially says that if it is not possible to select a solution that deletes a minimum number of vertices from  while simultaneously destroying all fragments in , then there is a bounded-size subset of fragments  that cannot all be destroyed by such a solution. The full importance of Lemma 1 will become clear in Section 4.

Organization

Section 2 provides basic preliminaries. In Section 3, we give some of the main ideas of the proof of Lemma 1. In Section 4 we show how Theorem 1 follows from a procedure that identifies relevant components. We give the procedure and its correctness proof later in the same section, while relying on Lemma 1. The proof of Lemma 1 is long and technical. In the appendix, we first develop a framework for boundaried labeled graphs and establish some useful auxiliary lemmata (Section B) and finally use these to prove the main lemma (Section C). Theorem 1 is proven in Section D in the appendix. The proofs of statements marked () can be found in the appendix, Section A.

2 Preliminaries

For a positive integer  we use  as a shorthand for . For a set , let to denote the set of all subsets of . All graphs we consider are finite, undirected, and simple. A graph  consists of a vertex set  and edge set . The open neighborhood of a vertex  is denoted . For a vertex set , its open neighborhood is . For an edge  in a graph , contracting  results in the graph  obtained from  by removing  and , and replacing them by a new vertex  with . For a vertex set , we use  to denote the graph obtained from  by deleting all vertices in  and their incident edges. The subgraph of  induced by vertex set  is denoted .

[treedepth] Treedepth is defined as follows. The trivial one-vertex graph has treedepth . The treedepth of a disconnected graph  with connected components  is . The treedepth of a connected graph  is .

[labeled graph] Let be a set. An -labeled graph is a graph together with label function , assigning a (potentially empty) subset of labels to each vertex in . The labeled graph  is -restricted if each vertex has at most labels.

If an edge  is contracted in a labeled graph to obtain a new vertex , then the labelset of  is defined as .

[minor model] A minor model of a graph in a graph is a mapping assigning a branch set to each vertex , such that:

  • is nonempty and connected for all ,

  • for all , and

  • if , then there exist and such that .

The third condition implies that one can find an edge mapping  such that:

  • For all , edge  has one endpoint in  and the other in .

We will often use the existence of this edge mapping in our proofs.

For  we define , and we define  as the range of the minor model. A minor model  of  in  is called minimal if no minor model  exists with .

[labeled minor model] A labeled minor model of an -labeled graph in an -labeled graph is a mapping  as in Definition 2, that additionally satisfies the following: for all and  there exists such that .

If  contains a (labeled) minor model of , then we say that  contains  as a (labeled) minor and denote this as . Observe that  contains  as a (labeled) minor if and only if  can be obtained from  by deleting edges and vertices (and potentially labels), and contracting edges.

[] Let  and  be unlabeled graphs, let , and let  be a minimal minor model of  in . Then  intersects at most  connected components of .

We denote the size of an optimal -Deletion solution on  by , and the set of optimal solutions by . In our bounds, we use the notation  for some identifier(s)  to denote a constant that only depends on .

[] Let  be a fixed set of (unlabeled) graphs, let  be a constant, and let  be a set. For any set  of -labeled graphs and host graph  with , one can:

  • compute  in  time;

  • determine whether there is a solution  such that  contains no graph from  as a labeled minor, in time  for some function .

Here  counts the number of elements of  that appear in the labelset of at least one vertex in at least one graph of .

3 Overview of the main lemma

In this section we discuss Lemma 1, whose long and technical proof is deferred to the appendix. The strength of the lemma comes from the fact that the bound on  is independent of the size of the graph  and of the number of labels  used on labelsets of vertices of .

Figure 1: Two constructions of graphs and sets for , where no optimal -deletion breaks , but for any there exists an optimal -deletion breaking . Top: any solution breaking both  and  (white vertices at the top) is larger than , but for any  there is a solution of size  breaking both  and  (white vertices at the bottom).

The statement of Lemma 1 is best-possible in several ways. First of all, the dependence of on instead of is essential. In Figure 1 (left), a construction of a graph of treewidth together with a set is shown. In this graph, no optimal -deletion (Vertex Cover) breaks all graphs in . However, for any there is an optimal vertex cover breaking . The example in Figure 1 can easily be extended to arbitrary , showing that there is a set with such that no optimal vertex cover breaks , yet there is no such that no optimal vertex cover breaks . Since is not bounded in terms of and , this shows that cannot be replaced by .

Secondly, the assumption that is -saturated cannot be avoided already for (corresponding to Feedback Vertex Set). In Figure 1 (right) we show an example of a graph of treedepth and a set of size that consist of single vertices of two labels each, where we again cannot properly bound the size of . The example is shown for but can easily be generalized to arbitrary , without increasing the treedepth. For any there exists an optimal -deletion breaking , while is not bounded in terms of and .

The proof of Lemma 1 follows an inductive strategy that mimics how a recursive algorithm would solve -Deletion on a bounded-treedepth graph . We pick a vertex  whose removal decreases the treedepth, and branch on whether  is part of the solution or not. If so, we remove  and recurse on a graph of smaller treedepth; if not, then we continue looking for solutions in which  is forbidden to be removed. The process builds up a set  with the property that removing  decreases the treedepth by , and we are only interested in solutions disjoint from . This proceeds while  remains connected; the branching depth is bounded since . When  becomes disconnected, we must take a more involved approach. We recurse on each of the connected components of  separately and find -Deletion solutions there. But solutions for different components of  may not combine into a solution for , since various fragments of -minors left behind in different components of , may be combined through their connections to  to form a forbidden minor. For this reason, when we recurse on connected components of  we place additional restrictions on the solutions chosen there, to ensure they also break fragments of -minors in such a way that the solutions can be properly combined.

Our approach to bound the size of  is built on top of this inductive strategy. While branching over various ways to form an -Deletion solution, we additionally branch on what fragments of labeled -minors are left behind by the solution in the various components of . By exploiting the saturatedness of  in a crucial way, we obtain the desired bound on . The formalization of these ideas requires an extensive theory of how fragments of a forbidden minor in various components of  may combine to form a forbidden minor in , which is developed in Appendix B.

4 Kernelization for -Deletion

In this section we describe the recursive approach to kernelize the -Deletion problem using a constant-treedepth modulator. The correctness of this strategy will crucially depend on Lemma 1. Lemma 4 identifies essential components in the input.

Let  be a finite set of connected graphs and let  be a constant. There is a polynomial-time algorithm that, given a graph  along with a modulator  such that , outputs an induced subgraph  of  together with an integer  such that  and  has at most  connected components.

Before proving this lemma, we show how it implies Theorem 1.

Theorem 1.

For every fixed finite set  of connected graphs and every constant , the -Deletion problem parameterized by a treedepth- modulator has a polynomial kernelization.

Proof.

Consider an input  to -Deletion. The proof is by induction on .

() If , then  is an independent set and any connected component of  contains one vertex. Apply Lemma 4 to find an induced subgraph  of  and integer  such that , which implies that  has answer yes if and only if  has answer yes. Now  has  single-vertex connected components. It follows that  has at most  vertices, which is polynomial in  for fixed . Hence  forms a polynomial kernel.

() For , we apply Lemma 4 on the input  and find  and  as above. We will augment the modulator  into a superset  to ensure that . To this end, we consider each connected component  of . If  consists of a single vertex then its treedepth is already smaller than . Otherwise,  is a connected graph with more than one vertex, and by Definition 2 there is a vertex  such that . Since the Treedepth problem parameterized by the target width is fixed-parameter tractable [39], and  is a constant, we can find such a vertex  by trying all options for  and computing the treewidth of the resulting graph in  time. (Alternatively, we can compute a treedepth-decomposition of  using the algorithm of Reidl et al. [39] and take its root as .) We initialize  as . For each component  of  with treedepth larger than one, we add the corresponding treedepth-decreasing vertex  to .

Since Lemma 4 guarantees that the number of connected components of  is polynomial in  for fixed  and , the resulting modulator  has size polynomial in . Moreover, it guarantees that . Hence we now have an instance  of -Deletion parameterized by a treedepth- modulator, with the same answer as . We apply the kernel for the parameterization by a treedepth-() modulator, which outputs an instance ) with the same answer as  and therefore as . By induction, the size of  is bounded by some polynomial in , which in turn is bounded by a polynomial in . Hence  has size  for some suitably chosen constant, and we output  as the result of the kernelization. ∎

Now we prove Lemma 4.

Proof of Lemma 4.

Let  be the connected components of . To reduce their number, we have a single reduction rule stated in terms of labeled graphs. With each connected component , we naturally associate an -labeled graph  by assigning a vertex  the labelset . We are interested in which of these labeled graphs have optimal -Deletion solutions that also hit certain fragments of potential -minor-models. We therefore define a set  which is a superset of the relevant fragments. We use  as a shorthand for . Let  consist of the connected -restricted -labeled graphs that have at most  edges. We consider two -labeled graphs to be identical if there is an isomorphism between them that respects the labelsets.

Claim .

.

Proof.

Graphs in  have at most  vertices. There are less than  distinct choices for the graph structure of a member of , since there are less than  different -vertex graphs. For each vertex, there are less than  choices for a labelset of size at most . Hence each graph structure  can appear with less than  different choices of labeling function, giving an overall bound  that is polynomial in . ∎

Choose  such that Lemma 1 guarantees that for this choice of  and the treedepth bound , one can always find  of size at most . Let , and . Consider the following marking procedure.

Procedure .

For each set  of size at most , do the following. Let

Mark  arbitrarily chosen components from , or mark all of them if there are fewer than .

Let  denote the marked components, , and let . The procedure can be executed in polynomial time, using variants of Courcelle’s theorem to find the sets . We explain how this is done in Lemma 2. Since , the number of subsets of  over which we iterate is polynomial in  and therefore in . Since the graphs in  are -restricted, the number of labels involved is constant for fixed  and , and therefore Lemma 2 guarantees a polynomial running time.

Claim .

.

Proof.

The procedure loops over  subsets . For each such set, we mark at most  components. ∎

The pair  is the desired outcome of Lemma 4. It remains to prove that . This follows from Claim 4 by induction.

Claim .

For any unmarked component .

Proof.

Let . Clearly, any solution for the graph  can be partitioned into a solution for  and a solution for , so that . We focus on proving the converse. Let  be an optimal solution on . Let  and let  contain those graphs for which the labelset of each vertex is contained in . Now define:

 there are fewer than components  of  (1)

Intuitively, one may think of  as those labeled graphs (that represent potential fragments of forbidden -minors) that can be realized in only few () components of  after removing the solution . When lifting the solution  in  to a solution in  by adding a solution in , it will be crucial to break all -labeled minor models of  in ; the fragments  that remain in many different components turn out to be irrelevant.

For a subset  of labels, let  be the labeled graph consisting of a single vertex with labelset . Let  and observe that . We prove:

(2)

Suppose  for suitable . Then there are  components of  that have  as labeled minor after removing the solution . Take  such components , and associate each one to a distinct vertex of . The fact that  is a labeled minor of  for each , implies that in each such component there is a connected vertex subset  such that each label of  appears at least once on a vertex of . Considering the corresponding vertex subset in  and taking into account that the labeling of  represents adjacency to  in , this implies that we can contract each  into a single vertex  that becomes adjacent to all vertices of . Then contract each  into a distinct vertex of : these minor operations on graph  turn  into a clique of size . Hence any graph on  vertices is a minor of , contradicting that  is -minor-free since  has a graph on  vertices. So (2) holds.

Now consider the unmarked component  in the statement of Claim 4, and consider its labeled version . We say that a vertex set  breaks the minor models of the -labeled graphs in , or simply breaks in , if does not contain any graph in as a labeled minor. We first show the following.

(3)

To establish (3), assume that no solution of size  in  breaks . We will use Lemma 1, together with our marking scheme, to argue for a contradiction. Observe that (2) implies that  is an -saturated set of -labeled graphs. If no optimal solution on  breaks , then by Lemma 1 there is a set  of size at most  such that no optimal solution on  breaks . Since the assumption that (3) does not hold means that the unmarked  was eligible to be marked for the set  in our procedure above, it has marked  other components  of . For each , there is no -Deletion solution of size  in  that breaks  in the labeled version . Since , by (1) we have for each graph  that there are fewer than  components  among  for which  contains  as a labeled minor. Since , it follows that there are at most  indices  for which  contains some graph from  as a labeled minor. But since , there are at least  components  in which all -minors are broken by . Since no optimal solution breaks  in the marked components, we have  for at least  components. But this contradicts that  is an optimal solution to -Deletion on : since  consists of connected graphs, we can form a solution  by taking  together with a set of size  from each component  of . Since  for all , with strict inequality for at least  components, we have . This contradicts that  is an optimal solution and establishes (3).

Hence there exists a solution in breaking of size . We prove:

(4)

This will complete the proof of Claim 4, since . Assume for a contradiction that  contains some graph  as a minor. Consider a minimal minor model of  in , which is given by a vertex mapping , and let  be a corresponding edge mapping.

Out of all possible minimal minor models of  in , select a model  that minimizes the quantity . Observe that if , then  is also a valid model in , contradicting that  is a solution to -Deletion on . So in the remainder we consider the case that the minor model contains at least one vertex of . We will build a minimal minor model of  in  using strictly fewer vertices of , thereby contradicting the choice of .

Figure 2: This figure shows how to define based on and , and how to modify the minor model of in such that it uses fewer vertices of , in the proof of (4) in Claim 4.

Consider the -labeled subgraph  of  obtained by the following procedure, which is illustrated in Figure 2:

  1. Start from the -labeled subgraph of  induced by , where each vertex  has labelset . As observed above, this subgraph is not empty.

  2. Remove all edges from this subgraph, except those in the range of  and those that connect two vertices that belong to a common branch set under .

  3. Contract every edge between two vertices that belong to a common branch set of , obtaining an -labeled graph . (Recall that labelsets merge during edge contraction.)

Observe that  has at most  edges, since each edge remaining in  corresponds to an edge in the range of . We claim that  is an -restricted graph: the labelset of each vertex has size less than . To see this, observe that if some vertex of  has a labelset  of size at least , then the pre-image of this vertex corresponds to a connected vertex subset  of  such that . Since  is a minor model in , this would imply that  has the one-vertex graph  with labelset  as a labeled minor. But  by (2), while  breaks all labeled -minors in  by definition; a contradiction. Hence  is indeed -restricted.

Let  be an arbitrary connected component of . Since  is connected, -restricted, and contains at most  edges, we have . As  clearly occurs as a labeled minor of , while  breaks  in , we have . By definition of , this implies there are at least  connected components  of  such that  contains  as -labeled minor for each . By Lemma 2, the minimal model  in  intersects at most  components of  and therefore of . Since  also intersects , it follows that some  is disjoint from the range of .

To finish the argument, fix  such that  and  contains  as -labeled minor. Let  denote the vertices of  whose contraction in the process above resulted in the connected component  of . Then it is straightforward to verify that  contains  as a minor. The role that vertices of  played in the minor model  can be replaced by the vertices of : each edge of  that was realized between vertices of  yielded an edge of  which is realized by a labeled -minor in ; each fragment of a branch set that was realized within  yielded a vertex of  that is realized in the -minor in ; and finally the connectivity of the branch sets is ensured because the labeling ensures that for all fragments of branch sets in  that were adjacent to vertices of , the branch set of the -minor in  realizing that fragment is also adjacent to all those vertices of . Hence there is a minimal -minor in  whose range is a subset of . Since  is not empty, this contradicts our choice of  as a minimal -model minimizing the intersection with . ∎

This concludes the proof of Lemma 4. ∎

5 Conclusion

Our goal in this paper was to obtain polynomial kernelizations for a wide range of graph problems, in terms of a rich class of structural parameterizations. We obtained polynomial kernelizations for -Deletion problems parameterized by a constant-treedepth modulator. The kernelization algorithm as presented here is only of theoretical interest. While the kernel size is polynomial for fixed  and , the degree of the polynomial grows very quickly with  and . It would be desirable to have a uniformly polynomial kernel size, of the form  for some constant  and function . Unfortunately, Theorem 1 shows that even for the simplest choice of , corresponding to the Vertex Cover problem, the degree of the polynomial must depend exponentially on  and no uniformly polynomial kernelization exists. The bad news also extends in the other direction: when taking the simplest choice for  and working with a treedepth-one modulator (a vertex cover), the degree of the polynomial in the kernel size for -Deletion must depend on  [23, Theorem 1.1] and a uniformly-polynomial kernel does not exist.

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Appendix A Omitted proofs from Section 2

Proof of Lemma 2.

Consider a minimal minor model  and let  be a corresponding edge mapping. For each , the graph  is connected by definition; let  be a spanning subtree of .

For each tree  that consists of more than one vertex, all leaves of  are incident on an edge in the range of : if  is a leaf of  not incident on an edge in the range of , then replacing  by  preserves connectivity of the branch set and validity of the edge mapping . This yields a minor model of  in  of smaller range, contradicting the minimality of .

We call a connected component  of  a terminal component for  if one of the following holds:

  • component  contains a vertex of  incident on an edge in the range of , or

  • is a single-vertex tree contained in  (which occurs when  is isolated in ).

A component  of  is a terminal component if it is a terminal component for some . Observe that an edge  cannot have endpoints in two different components of , as the presence of such an edge would mean that they are connected and form a single component. Hence each edge of  contributes at most one terminal component, implying that the total number of terminal components is at most .

Call a connected component  of  a nonterminal component for  if  contains a vertex of , but  is not a terminal component for . Intuitively, the minor model uses  to connect two vertices of . For  define . We bound the number of nonterminal components for  by , as follows.

Consider the graph  on vertex set  obtained from  by repeatedly contracting any edge that has at most one endpoint in , which is possible since  is connected. If  is a nonterminal component for , then each component of  has at least two -neighbors in  since  has no leaves in  by our observation above. Hence in the contraction process that turns  into , the contraction of a nonterminal component  contributes at least one edge to . No other component can contribute this same edge, as that would contradict the fact that  is acyclic. Hence the number of nonterminal components for  is bounded by the number of edges of . As any contraction of an acyclic graph is acyclic, it follows that  is an acyclic graph on vertex set . Hence it has at most  edges, yielding the desired bound on the number of nonterminal components for