1 Introduction
How, and under which circumstances, can a polynomialtime algorithm prune the easy parts of an NPhard 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 NPhard 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 polynomialtime 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 NPhard 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 nonexistence of polynomial kernelizations under complexitytheoretic 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 cubicvertex kernel. The result effectively conveys that large instances of Vertex Cover that are vertexdeletions away from being acyclic, can be shrunk to size in polynomial time.
Problem statement
To understand the power of polynomialtime preprocessing algorithms over inputs to NPhard 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 NPhard 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 metatheorems 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 MinorFree Deletion problems, henceforth abbreviated as Deletion problems, to capture a wide class of NPhard 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 solutionsize 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.^{1}^{1}1Bodlaender 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 vertexdeletion 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 vertexdeletion 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.^{2}^{2}2The 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 minorfree?
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 singleexponential 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 constanttreedepth 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 constanttreedepth 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 constanttreedepth 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 wellquasiordering, which are sources of algorithmic nonconstructivity 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 solutionsize 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 solutionsize 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 constanttreedepth 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 yesinstance if and only if is. This effectively shows that there is an optimal solution on in which the nonessential 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 straightforwardly from this pruning of nonessential 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 standalone 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 bestpossible. 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 boundedsize 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 onevertex 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 .
The statement of Lemma 1 is bestpossible 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 boundedtreedepth 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 constanttreedepth 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 polynomialtime 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 singlevertex 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 fixedparameter 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 treedepthdecomposition 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 treedepthdecreasing 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 minormodels. 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 minorfree 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 .
Consider the labeled subgraph of obtained by the following procedure, which is illustrated in Figure 2:

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

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 .

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 preimage of this vertex corresponds to a connected vertex subset of such that . Since is a minor model in , this would imply that has the onevertex 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 constanttreedepth 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 treedepthone modulator (a vertex cover), the degree of the polynomial in the kernel size for Deletion must depend on [23, Theorem 1.1] and a uniformlypolynomial 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 singlevertex 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