Keywords

1 Introduction

For NP-complete problems, a polynomial time algorithm solving any problem instance exactly is unlikely to exist. However, as one is often interested in solving specific instances, one can try to exploit characteristics of problem instances and develop algorithms that are fast when the input has certain properties. We therefore associate a parameter with each problem instance. In our context, a problem instance is a graph for which we ask for the existence of a vertex set of size at most \(\ell \) having certain properties. Such a parameterized instance can be denoted with a triple \((G, \ell , k)\), where we are asking for the existence of a solution of size at most \(\ell \) for a graph G with parameter k. We say that an algorithm is fixed parameter tractable (FPT) for such a parameterization if it solves any instance \((G, \ell , k)\) of size n, as described above, in time bounded by \(f(k) n^{\mathcal {O}(1)}\) for some computable function \(f :\mathbb {N}\rightarrow \mathbb {N}\).

A strongly related field is that of kernelization. This field focuses on reducing a parameterized instance \((G, \ell , k)\) in polynomial time to an equivalent instance \((G', \ell ', k')\) whose size is bounded by a computable function of the parameter. We speak of a polynomial kernel when this function is a polynomial. It is known that a decidable parameterized problem is fixed parameter tractable if and only if it admits a kernelization (cf. [12, Proposition 4.7.1]). In our quest for determining which parameterizations enable efficient algorithms, it is therefore interesting to determine those that allow a polynomial kernel.

This paper focuses on polynomial kernels for the undirected Feedback Vertex Set problem, which is an NP-complete problem in graph theory as originally identified by Karp [21]. For an undirected graph G, a vertex set \(X \subseteq V(G)\) is a feedback vertex set if the graph is acyclic after removal of X. We call a vertex set whose removal yields a graph in some graph class \(\mathcal G\) a \(\mathcal G\)-modulator and define the deletion distance to \(\mathcal G\) as its minimum size. The Feedback Vertex Set problem then asks for the minimum size of such a feedback vertex set, or equivalently, the deletion distance to a forest. For a graph G, we let \({\textsc {fvs}}(G)\) (the feedback vertex number of G) denote that minimum size. Our main question is for which parameterizations the Feedback Vertex Set problem admits a polynomial kernel.

Before exploring the Feedback Vertex Set problem further, we should mention the related Vertex Cover problem. It asks for a minimum set of vertices hitting all edges in a graph. While a kernel in the solution size with a linear number of vertices can be obtained using various techniques [1, 7,8,9, 24], a polynomial kernel in a structurally smaller parameter was only discovered in 2011, when Jansen and Bodlaender developed a polynomial kernel in the feedback vertex number of a graph [18]. From there, many polynomial kernels for Vertex Cover were discovered in modulators to even larger graph classes [4, 13, 15, 23]. In 2020, Bougeret, Jansen and Sau proved the following characterization under common hardness assumptions: Vertex Cover admits a polynomial kernel in the size of a modulator to a minor-closed graph family \(\mathcal G\) if and only if \(\mathcal G\) has bounded bridge-depth [3]. With this result, they generalized all existing work on kernels in the size of modulators to minor-closed graph families, and they proved that their results cannot be improved further under common hardness assumptions.

For Feedback Vertex Set, the first polynomial kernel with size bound \(\mathcal {O}(k^{11})\) was obtained in 2006 and it was subsequently improved to a quadratic kernel [2, 6, 26]. After the improvements for Vertex Cover, researchers also tried to develop polynomial kernels in smaller parameters for Feedback Vertex Set [17, 20, 22]. It remained an open problem whether these results could be generalized further or whether there exists some parameter that characterizes Feedback Vertex Set similarly to how bridge-depth characterizes Vertex Cover. In particular, Bougeret, Jansen and Sau suggested in their paper on Vertex Cover that the deletion distance to constant bridge-depth might also be an interesting parameter to consider for problems such as Feedback Vertex Set. We therefore aim to answer the question for which graph families \(\mathcal G\) the Feedback Vertex Set problem admits a polynomial kernel when parameterized by the size of a \(\mathcal G\)-modulator.

Our Results. To our initial surprise, the results for Vertex Cover cannot be generalized to Feedback Vertex Set. It turns out that a minor-closed graph family \(\mathcal G\) must have bounded elimination distance to a forest (Definition 1), in order to allow a polynomial kernel in a \(\mathcal G\)-modulator. This concept was introduced by Bulian and Dawar [5] and is another generalization of the more common parameter treedepth [25]. The elimination distance to a forest is the minimum number of rounds needed to transform the graph into a forest when removing one vertex from each connected component in each round. Our result is described in Theorem 1. Proofs of statements marked \((\star )\) are deferred to the full version [10].

Theorem 1

(\(\star \)). Assume \(\textrm{NP}\not \subseteq \textrm{coNP}/ \textrm{poly}\) and let \(\mathcal G\) be a minor-closed graph family. Then Feedback Vertex Set admits a polynomial kernel in the size of a \(\mathcal G\)-modulator if and only if \(\mathcal G\) has bounded elimination distance to a forest.

The minor-closed and hardness assumptions are only needed for the lower bound. To the best of our knowledge, our kernel generalizes all known polynomial kernels for the Feedback Vertex Set problem. Both the kernel and its correctness proof follow the structure of the kernel for \(\mathcal F\)-Minor Free Deletion in the deletion distance to a graph of constant treedepth by Jansen and Pieterse [20]. The correctness proof of their kernel crucially relies on their Lemma 3 whose technical proof spans thirty pages. We require a variation of this lemma. On the one hand, our variation is more involved since it deals with elimination distance to a forest rather than treedepth; on the other hand, it is simpler since it concerns only Feedback Vertex Set rather than \(\mathcal F\)-Minor Free Deletion. As a result of this simplification, we can formulate the lemma without the use of minors. Roughly speaking, the lemma says that in a graph G of bounded elimination distance to a forest, if no minimum feedback vertex set exists which simultaneously hits a prescribed set of partial cycles (single vertices in a set S or paths between two terminals in a set T), then the same holds for some sets \(S^* \subseteq S\) and \(T^* \subseteq T\) of constant size. As shown in previous work, this limited sensitivity with respect to whether optimal solutions can break all partial forbidden structures is crucial for the kernelization complexity. As one of our main contributions, we prove this lemma using a strategy that differs significantly from the one followed in earlier work [20].

Lemma 1

(\(\star \)). Let G be a connected graph with disjoint vertex sets \(S,T \subseteq V(G)\). Suppose that any minimum feedback vertex set X of G misses some vertex from S or leaves two vertices from T connected in \(G-X\). Then there exist sets \(S^* \subseteq S\) and \(T^* \subseteq T\) whose sizes only depend on the elimination distance to a forest of G, such that any minimum feedback vertex set X of G misses some vertex from \(S^*\) or leaves two vertices from \(T^*\) connected in \(G-X\).

Once Lemma 1 is proven, the kernelization upper bounds follow similarly to earlier work [20]. As for the lower bound in Theorem 1, we are also able to generalize our proof for other \(\mathcal F\)-Minor Free Deletion problems as described in Theorem 2.

Theorem 2

(\(\star \)). Let \(\mathcal G\) be a minor-closed family of graphs and let \(\mathcal F\) be a finite set of biconnected planar graphs on at least three vertices. If \(\mathcal G\) has unbounded elimination distance to an \(\mathcal F\)-minor free graph, then \(\mathcal F\)-Minor Free Deletion does not admit a polynomial kernel in the size of a \(\mathcal G\)-modulator, unless \(\textrm{NP}\subseteq \textrm{coNP}/ \textrm{poly}\).

Organization. Section 2 introduces all relevant terminology. Section 3 presents our kernel and thereby proves the ‘if’ direction of Theorem 1. Then Sect. 4 contains the proof of Theorem 2, thereby also proving the ‘only if’ direction of Theorem 1. Lastly, Sect. 5 contains our conclusions and discusses future work.

2 Preliminaries

For a positive integer n, we use the shorthand [n] for the set of all natural numbers i with \(1 \le i \le n\). All graphs we consider are finite, undirected and simple. When G is a graph, we let V(G) denote the vertex set of G and E(G) the edge set. For \(S \subseteq V(G)\), the graph \(G-S\) is the graph where all vertices in S and all incident edges are removed, and the graph G[S] is the subgraph of G induced by the vertices in S. When an edge exists between two vertices in G, we say that these vertices are adjacent. The neighbors of v in G, denoted with \(N_G(v)\), are the vertices adjacent to a vertex \(v \in V(G)\) in G. For \(S \subseteq V(G)\), we say that \(v \in V(G-S)\) is adjacent to S if there exists some edge between v and a vertex in S. The set \(N_G(S)\) contains all vertices \(v \in V(G - S)\) for which this holds. We will sometimes slightly abuse notation and speak of a vertex being adjacent to some subgraph, rather than to the vertices in that subgraph. We say that two vertices are connected in G when they are in the same connected component. The set \({\textsc {cc}}(G)\) denotes the set of connected components (or shortly components) of G. For sets \(S, T \subseteq V(G)\), we say that S separates T if each component of \(G-S\) contains at most one vertex from T. Notice that we do not require S and T to be disjoint. Such a set S is a vertex multiway cut of T in G. We use the notation \(\mathcal {O}_{\eta }(f(n))\) to describe the functions in \(\eta \) and n which can be bounded by \(g(\eta ) \cdot f(n)\) for some computable function g.

A concept that will be used extensively is the concept of graph minors. This uses the notion of edge contraction. When uv is an edge in a graph G, contracting this edge replaces vertices u and v by a new vertex whose set of neighbors is . Now H is a minor of G if H can be obtained from G by removing vertices, removing edges and contracting edges. Alternatively, one can define H to be a minor of G if there exists a minor model \(\phi :V(H) \rightarrow 2^{V(G)}\), such that for any \(v \in V(H)\) the graph \(G[\phi (u)]\) is connected, for any distinct \(u, v \in V(H)\) we have \(\phi (u) \cap \phi (v) = \emptyset \), and for any edge uv in H there exists an edge between a vertex in \(\phi (u)\) and a vertex in \(\phi (v)\) in G.

A graph G has an \(\mathcal H\)-minor for a set of graphs \(\mathcal H\) if G contains some graph \(H \in \mathcal H\) as a minor. For a minor model \(\phi \) of H in G and a set \(S \subseteq V(H)\), we use the shorthand notation \(\phi (S) {:}{=}\bigcup _{v \in S} \phi (v)\). We say that a minor model \(\phi \) of H in G is minimal, if there does not exist a minor model \(\phi '\) of H in G with \(\phi '(V(H)) \subsetneq \phi (V(H))\). A minor model \(\phi \) of H in G and a minor model \(\phi '\) of \(H'\) in G intersect if \(\phi (V(H)) \cap \phi '(V(H')) \ne \emptyset \).

2.1 Elimination Distance

Our work relies crucially on the concept of elimination distance as introduced by Bulian and Dawar [5].

Definition 1

(Elimination distance). Let G be a graph and \(\mathcal G\) a graph family. Then the elimination distance of G to \(\mathcal G\) is

figure b

We only consider graph families \(\mathcal G\) that are minor-closed. We use the shorthand \(\mathcal G_F\) for the graph family containing precisely all forests. We say that a graph family \(\mathcal G\) has bounded elimination distance to some graph class \(\mathcal H\) if there is a constant \(c \in \mathbb {N}\) such that all graphs \(G \in \mathcal G\) satisfy \({\textsc {ed}}_{\mathcal H}(G) \le c\). The elimination distance of a graph G to the empty graph is called the treedepth of G and is denoted with \({\textsc {td}}(G)\). More intuitively, the elimination distance to a graph class \(\mathcal G\) can be interpreted as the minimum number of ‘elimination iterations’ that are necessary to obtain a graph where every connected component is in \(\mathcal G\). In such an iteration, one is allowed to remove one vertex from each connected component. This interpretation leads to the notion of an elimination forest.

Definition 2

(\(\mathcal G\)-elimination forest). Let G be a graph and \(\mathcal G\) a graph family. A \(\mathcal G\)-elimination forest of G is a tuple \(\left( F, (B_u)_{u \in V(F)}\right) \) where F is a rooted forest and where each vertex \(v \in V(F)\) has a bag \(B_v \subseteq V(G)\) such that:

  • The bags define a partition of V(G), i.e. for any vertex \(v \in V(G)\) there is a unique vertex \(u \in V(F)\) with \(v \in B_u\).

  • For any non-leaf u of F, its bag \(B_u\) contains precisely one vertex.

  • For any leaf u of F, the graph \(G[B_u]\) is connected and \(G[B_u] \in \mathcal G\).

  • For any edge uv in G, let \(s, t \in V(F)\) be the vertices such that \(u \in B_s\) and \(v \in B_t\). Then s is an ancestor of t, or t is an ancestor of s in F.

We define the height of an elimination forest F to be the maximum number of edges on a path from the root to a leaf in F. One can prove with induction that this height is equal to the elimination distance we defined earlier.

We will use these elimination forests extensively for our kernel and therefore introduce some shorthand notation. Let \((F, (B_u)_{u \in V(F)})\) be a \(\mathcal G\)-elimination forest. Let v be a vertex in F. The tail of v, denoted with \({\textsc {tail}}(v)\), is defined as the union of \(B_u\) over all proper ancestors u of v. The closed tail \({\textsc {tail}}[v]\) also includes \(B_v\). Similarly, \({\textsc {tree}}(v)\) denotes the union of \(B_u\) over all proper descendants u of v and \({\textsc {tree}}[v]\) also includes \(B_v\). The subgraph of G induced by all vertices in \({\textsc {tree}}[v]\) is denoted with \(G_v\). We will sometimes slightly abuse notation and use \(G_v\) as a vertex set. We use the shorthand \(G_v^+\) to describe the induced subgraph on the vertices in \({\textsc {tree}}[v] \cup {\textsc {tail}}[v]\).

We will also introduce the notion of bridge-depth as introduced by Bougeret, Jansen and Sau [3]. A bridge in a graph G is an edge whose removal increases the number of connected components of G. The concept of bridge-depth now allows us to delete a set of vertices S as long as G[S] is connected and each edge in G[S] is a bridge in G. Such a structure G[S] is called a tree of bridges. Observe that a single vertex is always a tree of bridges.

Definition 3

(Bridge-depth). Let G be a graph. The bridge-depth of G is defined as

figure c

Cf. [3] for equivalent definitions. Lastly, we sometimes use the more common concept of treewidth. The treewidth of a graph G is denoted with \({\textsc {tw}}(G)\). We mention some useful properties of these concepts in Proposition 1.

Proposition 1

(\(\star \)). Let G be a graph with \(\mathcal G_F\)-elimination forest \(\left( F, (B_u)_{u \in V(F)}\right) \) and let \(\eta \) be an integer such that \({\textsc {ed}}_{\mathcal G_F}(G) \le \eta \). Let X be a minimum feedback vertex set in G and let v be a leaf in F. Then the following claims hold.

  1. 1.

    \({\textsc {tw}}(G) \le {\textsc {bd}}(G) \le {\textsc {ed}}_{\mathcal G_F}(G) + 1\).

  2. 2.

    X contains at most \(\eta \) vertices from \(B_v\).

  3. 3.

    If there exists a path in G from a vertex in \(B_v\) to a vertex outside \(B_v\), then this path contains a vertex in \({\textsc {tail}}(v)\).

  4. 4.

    Let \(u \in V(F)\), then u has at most \(\eta \) children c where \(X \,\cap \, G_c\) is not a minimum feedback vertex set in \(G_c\).

3 Kernelization Upper Bounds

Our kernel follows the structure of the polynomial kernel for \(\mathcal F\)-Minor Free Deletion when parameterized by a treedepth-\(\eta \) modulator for some integer \(\eta \) [20]. Our kernel relies crucially on the reduction rule specified in Lemma 2.

Lemma 2

( \(\star \) Cf.  [20], Lemma 6). There is a polynomial-time algorithm that, given a graph G with modulator \(X \subseteq V(G)\) such that \({\textsc {ed}}_{\mathcal G_F}(G - X) \le \eta \) for a constant \(\eta \), outputs an induced subgraph \(G'\) of G together with an integer \(\varDelta \) such that \({\textsc {fvs}}(G) = {\textsc {fvs}}(G') + \varDelta \) and \(G' - X\) has at most \(| X |^{\mathcal {O}_\eta (1)}\) components.

We can use this reduction rule to obtain a graph \(G'\) where \(G' - X\) has a bounded number of connected components. We can then identify a set of vertices \(Y \subseteq V(G'-X)\) with \(| Y | \le | X |^{\mathcal {O}_\eta (1)}\) such that \({\textsc {ed}}_{\mathcal G_F}(G' - X - Y) < \eta \). By definition of elimination distance, every connected component C of \(G'-X\) contains a vertex whose removal decreases \({\textsc {ed}}_{\mathcal G_F}(C)\). As we limited the number of connected components by applying Lemma 2, these vertices constitute a suitable set Y. Now observe that \(X \cup Y\) is a modulator to a graph with elimination distance to a forest \(\eta - 1\) and that \(| X \cup Y |\) is bounded by a polynomial in \(| X |\). One can therefore provide an inductive argument which repeatedly applies Lemma 2 and increases the modulator such that the elimination distance to a forest of the remaining graph decreases every iteration. Once we obtain a modulator to a graph with elimination distance to a forest 1 (which is a forest), we can apply a known polynomial kernel in the size of a feedback vertex set [16].

The reduction rule of Lemma 2 follows from the following key lemma using known techniques. We therefore focus our attention on the proof of Lemma 1.

Lemma 1

(\(\star \)). Let G be a connected graph with disjoint vertex sets \(S,T \subseteq V(G)\). Suppose that any minimum feedback vertex set X of G misses some vertex from S or leaves two vertices from T connected in \(G-X\). Then there exist sets \(S^* \subseteq S\) and \(T^* \subseteq T\) whose sizes only depend on the elimination distance to a forest of G, such that any minimum feedback vertex set X of G misses some vertex from \(S^*\) or leaves two vertices from \(T^*\) connected in \(G-X\).

We can split up Lemma 1 into two parts. Lemma 3 will bound the number of vertices in the \(\mathcal G_F\)-elimination tree that contain a vertex in S or T. This part corresponds to the original treedepth formulation in [20, Lemma 3], but is significantly simplified for our restricted setting. Lemma 4 bounds the number of vertices in S and T in a bag of the elimination tree. This covers the generalization to elimination distance to a forest and concludes the proof of Lemma 1. In the full version [10], we show how these lemmas imply Lemma 1.

Lemma 3

Let G be a connected graph with disjoint vertex sets \(S,T \subseteq V(G)\). Let \(\left( R, (B_u)_{u \in V(R)}\right) \) be a \(\mathcal G_F\)-elimination tree of G of height \(\eta \). Suppose that any minimum feedback vertex set X of G misses a vertex from S or leaves two vertices from T connected. Then this also holds for some subsets \(S^* \subseteq S\) and \(T^* \subseteq T\), such that any vertex in the elimination tree has at most \(3\eta \cdot 2^\eta \) children u for which \(G_u\) contains a vertex from \(S^*\) or \(T^*\).

Proof

In analogy to the original formulation in [20], a labeled vertex is a vertex in S or T. When we remove a label from a vertex, we remove the vertex from S and T while the vertex remains in the graph. Suppose that we pick \(S^*\) and \(T^*\) such that no minimum feedback vertex set contains \(S^*\) and separates \(T^*\), while the latter property does not hold for any other pair of sets \(S', T'\) with \(S' \subseteq S^*\) and \(T' \subseteq T^*\). We claim that for such sets \(S^*\) and \(T^*\), any vertex in the elimination tree R has at most \(3\eta \cdot 2^\eta \) children u for which \(G_u\) contains a labeled vertex. We will also refer to the set \(T^*\) as the set of terminals.

Assume towards a contradiction that vertex v has more child subtrees with labels. Let these children be \(c_1, \dots , c_\ell \). For each of these children \(c_i\), there exists a minimum feedback vertex set \(X_i\) in G that witnesses the fact that the labels cannot be removed from \(G_{c_i}\). This set \(X_i\) will therefore miss a vertex in \(S^* \cap G_{c_i}\) or leave a vertex in \(T^* \cap G_{c_i}\) connected to some other vertex in \(T^*\), while \(X_i\) contains all vertices in \(S^* {\setminus } G_{c_i}\) and separates all vertices in \(T^* {\setminus } G_{c_i}\). Define \(Z_i {:}{=}{\textsc {tail}}[v] {\setminus } X_i\).

Now fix a set \(Z \subseteq {\textsc {tail}}[v]\). We will bound the number of children \(c_i\) for which \(Z_i = Z\) by \(3 \eta \). Suppose towards a contradiction that there are \(3 \eta + 1\) of these children. Let C be the set containing these vertices. Pick some child \(c_j \in C\) and observe the following.

  • By Proposition 1.4, there are at most \(\eta \) children \(c_i \in C\) where \(X_j \cap G_{c_i}\) is not a minimum feedback vertex set in \(G_{c_i}\).

  • There are at most \(\eta \) children \(c_i \in C\) with \(i \ne j\) such that a terminal in \(G_{c_i}\) is connected to a vertex in Z in \(G^+_{c_i} - X_j\), i.e. (recall the notation in Sect. 2.1) in the induced subgraph on the remaining vertices in the subtree and tail of \(c_i\). Otherwise, two children other than \(c_j\) have a terminal connected to the same vertex in Z, while \(X_j\) separates all terminals outside \(G_{c_j}\).

  • There are at most \(\eta - 1\) children \(c_i \in C\) such that in \(G_{c_i}^+ - X_j\), there exists a path between distinct vertices in Z that uses some vertex in \(G_{c_i}\). Otherwise, we claim that we can directly construct a cycle in \(G - X_j\). Consider for example the auxiliary (multi)graph on vertex set Z which contains, for each child \(c_i \in C\) for which \(G_{c_i}^+ - X_j\) contains such a path, say between \(z_1, z_2 \in Z\), one edge \(z_1 z_2\). This auxiliary graph contains a cycle since it has too many edges to be acyclic, which implies that there exists a cycle in \(G - X_j\).

Pick a child \(c_k \in C\) which is neither \(c_j\) nor in the list of \(3\eta -1 \) children above. As \(| C | > 3\eta \), such a vertex exists. By the first item above, we can deduce that \(X_j \cap G_{c_k}\) is a minimum feedback vertex set in \(G_{c_k}\). Besides, this set contains \(S^* \cap G_{c_k}\), it separates all terminals in \(T^* \cap G_{c_k}\), and it separates \(T^* \cap G_{c_k}\) from Z. Furthermore, no path exists in \(G_{c_k}^+ - X_j\) that connects two vertices in Z and also contains some vertex in \(G_{c_k}\).

Claim 1

(\(\star \)). The set \(X' {:}{=}(X_k {\setminus } G_{c_k}) \cup (X_j \cap G_{c_k})\) is a minimum feedback vertex set in G which contains \(S^*\) and separates \(T^*\).

Claim 1 contradicts that any minimum feedback vertex set in G misses a vertex in \(S^*\) or leaves two vertices in \(T^*\) connected. We conclude that there are at most \(3\eta \) children \(c_i\) of v for which a witnessing minimum feedback vertex set has \(Z_i = Z\). As there are at most \(2^\eta \) subsets of \({\textsc {tail}}[v]\) for any non-leaf v, this leads to the bound of at most \(3 \eta \cdot 2^\eta \) children for which the labels cannot be removed.    \(\square \)

Lemma 4

Let G be a connected graph with disjoint vertex sets \(S,T \subseteq V(G)\). Let \(\left( R, (B_u)_{u \in V(R)}\right) \) be a \(\mathcal G_F\)-elimination tree of G of height \(\eta \). Suppose that any minimum feedback vertex set X of G misses a vertex from S or leaves two vertices from T connected. Then this also holds for some subsets \(S^* \subseteq S\) and \(T^* \subseteq T\), such that for any leaf u in the elimination tree, the set \(B_u\) contains at most \(\eta \) vertices from \(S^*\) and at most \(\mathcal {O}(\eta ^2)\) from \(T^*\).

Proof

Pick some leaf v of elimination tree R, for which we want to ensure that there are \(\mathcal {O}(\eta ^2)\) vertices with labels among vertices in \(Y {:}{=}B_v\). Define \(S_Y {:}{=}S \cap Y\) and \(T_Y {:}{=}T \cap Y\). Our goal is to obtain subsets \(S_Y^* \subseteq S_Y\) and \(T_Y^* \subseteq T_Y\) whose sizes are \(\mathcal {O}(\eta ^2)\), such that every minimum feedback vertex set misses a vertex from \(S_Y^* \cup (S {\setminus } Y)\) or leaves a pair of terminals in \(T_Y^* \cup (T {\setminus } Y)\) connected. By applying this operation to all leaves of the elimination tree, we obtain the sets promised by Lemma 4.

The construction of \(S_Y^*\) is straightforward. If \(| S_Y | > \eta + 1\), we let \(S_Y^*\) be an arbitrary subset of \(S_Y\) of size \(\eta + 1\). Otherwise, \(S_Y^* = S_Y\).

Claim 2

Let X be a minimum feedback vertex set in G. If X misses a vertex in S, then it also misses a vertex in \(S_Y^* \cup (S {\setminus } Y)\).

Proof

If X misses a vertex in \(S {\setminus } Y\), then the implication is trivial. Therefore assume X misses a vertex in \(S_Y\). If this vertex is not in \(S_Y^*\), then \(| S_Y^* | = \eta + 1\) by construction. By Proposition 1.2, we know that \(| X \cap S_Y^* | \le \eta \) so X misses a vertex in \(S_Y^*\).    \(\blacksquare \)

For the construction of \(T_Y^*\) we distinguish two cases. First, we assume that \(T_Y\) cannot be separated with \(\eta \) vertices in G[Y] and make the following observation.

Proposition 2

(\(\star \)). Let G be a tree and \(T \subseteq V(G)\). If T cannot be separated with \(\eta \) vertices, then there exist \(\eta + 1\) vertex-disjoint paths whose endpoints are distinct vertices in T.

We define \(T_Y^*\) by taking the \(2\eta + 2\) endpoints of the paths guaranteed by Proposition 2. Observe that these vertices are all in \(T_Y\).

Claim 3

Suppose that \(T_Y\) cannot be separated with \(\eta \) vertices in G[Y]. Let X be a minimum feedback vertex set in G. Then X leaves two vertices in \(T_Y^*\) connected.

Proof

By Proposition 1.2, X can only intersect \(\eta \) of the \(\eta + 1\) vertex-disjoint paths that were obtained through Proposition 2. Therefore, at least one path is disjoint from X, so its endpoints in \(T_Y^*\) are connected in \(G-X\).    \(\blacksquare \)

It remains to consider the case where \(T_Y\) can be separated with \(\eta \) vertices. Let Z be a vertex multiway cut of \(T_Y\) in G[Y] with \(| Z | \le \eta \) and let \(\mathcal C {:}{=}{\textsc {cc}}(G[Y] - Z)\). Observe that each of these connected components is a tree with at most one vertex in \(T_Y\). Let \(\mathcal C_T \subseteq \mathcal C\) be the set of components that contain a vertex in \(T_Y\). We are now going to mark components. For each \(z \in Z\), mark \(\eta + 2\) components in \(\mathcal C_T\) that are adjacent to z in G[Y], or all if there are fewer. Similarly, for each \(u \in {\textsc {tail}}(v)\) we mark up to \(\eta + 2\) components in \(\mathcal C_T\) that are adjacent to v in \(G_v^+\). Then we define \(T_Y^*\) to be the union of all vertices in \(T_Y\) in the marked components, together with \(Z \cap T_Y\). These are at most \(\eta (\eta \,+\, 2) \,+\, \eta (\eta + 2) \,+\, \eta \,=\, \mathcal {O}(\eta ^2)\) vertices.

Claim 4

Suppose that \(T_Y\) can be separated with \(\eta \) vertices in G[Y]. Let X be a minimum feedback vertex set in G and suppose that X leaves two vertices in T connected. Then X also leaves two vertices in \(T_Y^* \cup (T {\setminus } Y)\) connected.

Proof

Let Z be the vertex multiway cut used in the construction of \(T_Y^*\) and let \(t_1, t_2 \in T\) be two terminals that are connected in \(G - X\). If they are both in \(T_Y^* \cup (T {\setminus } Y)\), then the implication is trivial, so assume that \(t_1 \in T_Y\) but not in \(T_Y^*\). Observe that therefore \(t_1 \not \in Z\). Let P be a path from \(t_1\) to \(t_2\) in \(G - X\) and let z be the first vertex on this path that is not in \(G[Y] - Z\). We now distinguish two cases. If \(z \in Z\), then observe that \(t_1\) was in a component in \(\mathcal C_T\) that was not marked. Then there are \(\eta + 2\) marked components in \(\mathcal C_T\) adjacent to z in G[Y] of which the terminals are in \(T_Y^*\). Only \(\eta \) of these components can be intersected by X by Proposition 1.2, so there exists a path between two terminals in \(T_Y^*\) in G[Y]. If \(z \not \in Z\), then we obtain that \(z \in {\textsc {tail}}(v)\) by Proposition 1.3 and the case follows analogously.    \(\blacksquare \)

This concludes the construction of the sets \(S_Y^*\) and \(T_Y^*\). If any minimum feedback vertex set in G misses a vertex in S or leaves a pair of terminals in T connected, then it also misses a vertex in \(S_Y^* \cup (S {\setminus } Y)\) or leaves a pair of terminals in \(T_Y^* \cup (T {\setminus } Y)\) connected. By applying this operation to all leaves of the elimination tree, we obtain the promised sets \(S^*\) and \(T^*\) which concludes the proof of Lemma 4.    \(\square \)

With Lemma 3 and Lemma 4 proven, we conclude the proof of Lemma 1: if any minimum feedback vertex set in a graph G misses a vertex from a set \(S \subseteq V(G)\) or leaves two terminals in a set \(T \subseteq V(G)\) connected, then this property also holds for sets \(S^* \subseteq S\) and \(T^* \subseteq T\) whose sizes only depend on \({\textsc {ed}}_{\mathcal G_F}(G)\). This is the key ingredient for the proof of Lemma 2, which leads to the kernel upper bound.

4 Kernelization Lower Bounds

In this section we summarize the main ideas behind the lower bound. We first introduce the notion of a necklace, which turns out to be a crucial structure.

Definition 4

Let G be a graph and let \(\mathcal F\) be a collection of connected graphs. G is an \(\mathcal F\)-necklace of length t if there exists a partition of V(G) into \(S_1, \dots , S_t\) such that

  • \(G[S_i] \in \mathcal F\) for each \(i \in [t]\) (these subgraphs are the beads of the necklace),

  • G has precisely one edge between \(S_i\) and \(S_{i+1}\) for each \(i \in [t-1]\),

  • G has no edges between any other pair of sets \(S_i\) and \(S_j\).

When the length of the necklace is not relevant, we simply speak of an \(\mathcal F\)-necklace. The following definition specifies a special type of necklace.

Definition 5

Let \(\mathcal F\) be a collection of connected graphs. Let G be an \(\mathcal F\)-necklace of length t. We say that G is a uniform necklace if it satisfies two additional conditions.

  • There exists a graph \(H \in \mathcal F\) such that each bead \(G[S_i]\) is isomorphic to H.

  • There exist \(x, y \in V(H)\) and graph isomorphisms \(f_i :V(H) \rightarrow V(G[S_i])\) for each bead \(G[S_i]\), such that for each \(i \in [t-1]\), the edge between \(G[S_i]\) and \(G[S_{i+1}]\) has precisely the endpoints \(f_i(x)\) and \(f_{i+1}(y)\).

These concepts are used to derive the following characterization. We say that a set contains arbitrarily long necklaces if there does not exist a constant c such that each necklace in the set has length at most c.

Lemma 5

(\(\star \)). Let \(\mathcal F\) be a finite collection of connected planar graphs. Any minor-closed graph family \(\mathcal G\) with unbounded elimination distance to an \(\mathcal F\)-minor free graph contains arbitrarily long uniform \(\mathcal F\)-necklaces.

Then we will prove the following lemma by giving a reduction from CNF Satisfiability parameterized by the number of variables [11].

Lemma 6

(\(\star \)). Let \(\mathcal F\) be a finite set of biconnected planar graphs on at least three vertices and let \(\mathcal G\) be a minor-closed graph family. If \(\mathcal G\) contains arbitrarily long uniform \(\mathcal F\)-necklaces, then \(\mathcal F\)-Minor Free Deletion does not admit a polynomial kernel in the size of a \(\mathcal G\)-modulator, unless \(\textrm{NP}\subseteq \textrm{coNP}/ \textrm{poly}\).

Lemma 5 and Lemma 6 together directly imply Theorem 2. We will explain the main ideas of the proof of Lemma 5 here. Our proof follows the proof by Bougeret et al. when they characterize graph families with unbounded bridge-depth [3]. Similar to their work, we define \({{\textsc {nm}}}_{\mathcal F}(G)\) to be the length of the longest \(\mathcal F\)-necklace that a graph G contains as a minor for a family of connected graphs \(\mathcal F\). Our goal is now to prove the existence of a small set X such that \({{\textsc {nm}}}_{\mathcal F}(G - X) < {{\textsc {nm}}}_{\mathcal F}(G)\) as described in Lemma 7.

Lemma 7

(\(\star \)). Let \(\mathcal F\) be a collection of connected planar graphs. Then there exists a polynomial function \(f_{\mathcal F} :\mathbb {N}\rightarrow \mathbb {N}\) such that for any connected graph G with \({{\textsc {nm}}}_{\mathcal F}(G) = t\), there exists a set \(X \subseteq V(G)\) with \(| X | \le f_{\mathcal F}(t)\) such that \({{\textsc {nm}}}_{\mathcal F}(G - X) < t\).

Bougeret et al. showed that one can derive a bounding function when the considered structures satisfy the Erdős-Pósa property [3]. This also is the case for \(\mathcal F\)-necklaces when the graphs in \(\mathcal F\) are connected and planar, so this approach would be suitable for our purposes as well. To derive a polynomial bound on the size of X, we use a different argument that uses treewidth and grid minors. We start with the following property of planar graphs.

Proposition 3

(\(\star \)). Any planar graph G on n vertices is a minor of the \(4n \times 4n\) grid.

Together with the Excluded Grid Theorem, this leads to the following treewidth bound.

Lemma 8

(\(\star \)). Let \(\mathcal F\) be a collection of connected planar graphs of at most n vertices each. There exists a polynomial \(f :\mathbb {N}\rightarrow \mathbb {N}\) with \(f(g) = \mathcal {O}(g^{19} \text {poly} \log g)\) such that for any graph G with \({{\textsc {nm}}}_{\mathcal F}(G) = t\), it holds that \({\textsc {tw}}(G) < f(4n(t+1))\).

To use this treewidth bound, we need a property similar to [3, Lemma 4.6].

Proposition 4

(\(\star \)). For any family of connected graphs \(\mathcal F\) and connected graph G with \({{\textsc {nm}}}_{\mathcal F}(G) > 0\), any pair of minor models of \(\mathcal F\)-necklaces of length \({{\textsc {nm}}}_{\mathcal F}(G)\) in G must intersect.

Proposition 4 is a generalization of the idea that in any connected graph, two paths of maximum length must intersect at a vertex. Given a graph G with a tree decomposition, we can use this property to identify a vertex in the tree decomposition such that the removal of all vertices in its bag decreases \({{\textsc {nm}}}_{\mathcal F}(G)\). This result is described in Lemma 9.

Lemma 9

(\(\star \)). Let \(\mathcal F\) be a collection of connected graphs. Let G be a connected graph with \({\textsc {tw}}(G) = w\) and \({{\textsc {nm}}}_{\mathcal F}(G) = t\). Then there exists a set \(Z \subseteq V(G)\) with \(| Z | \le w+1\) such that \({{\textsc {nm}}}_{\mathcal F}(G-Z) < t\).

The proof of Lemma 7 follows directly by combining Lemma 8 and Lemma 9. An inductive argument, analogous to [3, Theorem 4.8], remains to use this result to prove Lemma 5.

5 Conclusion and Discussion

We conclude that the elimination distance to a forest characterizes the Feedback Vertex Set problem in terms of polynomial kernelization. For a minor-closed graph family \(\mathcal G\), the problem admits a polynomial kernel in the size of a \(\mathcal G\)-modulator if and only if \(\mathcal G\) has bounded elimination distance to a forest, assuming \(\textrm{NP}\not \subseteq \textrm{coNP}/ \textrm{poly}\). In particular, this implies that Feedback Vertex Set does not admit a polynomial kernel in the deletion distance to a graph of constant bridge-depth under the mentioned hardness assumption. We also generalize the lower bound to other \(\mathcal F\)-Minor Free Deletion problems where \(\mathcal F\) contains only biconnected planar graphs on at least three vertices. It remains unknown whether such a lower bound also generalizes to collections of graphs \(\mathcal F\) that contain non-planar graphs.

An interesting open problem is whether similar polynomial kernels can be obtained for other \(\mathcal F\)-Minor Free Deletion problems. Regarding the field of fixed parameter tractable algorithms, it was recently shown [19] that for any set \(\mathcal {F}\) of connected graphs, \(\mathcal F\)-Minor Free Deletion admits an FPT algorithm when parameterized by the elimination distance to an \(\mathcal F\)-minor free graph (or even \(\mathcal H\)-treewidth when \(\mathcal H\) is the class of \(\mathcal F\)-minor free graphs). This generalizes known FPT algorithms for the natural parameterization by solution size. Regarding polynomial kernels, \(\mathcal F\)-Minor Free Deletion problems admit a polynomial kernel in the solution size when \(\mathcal F\) contains a planar graph [14]. Do polynomial kernels exist when the problem is parameterized by a modulator to a graph of constant elimination distance to being \(\mathcal {F}\)-minor free?