A Turing Kernelization Dichotomy for Structural Parameterizations of $\mathcal {F}$-Minor-Free Deletion

Abstract

For a fixed finite family of graphs $\mathcal {F}$, the $\mathcal {F}$-Minor-Free Deletion problem takes as input a graph G and an integer $\ell$ and asks whether there exists a set $X \subseteq V(G)$ of size at most $\ell$ such that $G-X$ is $\mathcal {F}$-minor-free. For $\mathcal {F} =\{K_2\}$ and $\mathcal {F} =\{K_3\}$ this encodes Vertex Cover and Feedback Vertex Set respectively. When parameterized by the feedback vertex number of G these two problems are known to admit a polynomial kernelization. Such a polynomial kernelization also exists for any $\mathcal {F}$ containing a planar graph but no forests.

In this paper we show that $\mathcal {F}$-Minor-Free Deletion parameterized by the feedback vertex number is $\mathsf {MK[2]}$-hard for $\mathcal {F} = \{P_3\}$. This rules out the existence of a polynomial kernel assuming $\mathsf {NP}\not \subseteq \mathsf {coNP/poly}$, and also gives evidence that the problem does not admit a polynomial Turing kernel. Our hardness result generalizes to any $\mathcal {F}$ not containing a $P_3$-subgraph-free graph, using as parameter the vertex-deletion distance to treewidth $\min {{\,\mathrm{tw}\,}} (\mathcal {F})$, where $\min {{\,\mathrm{tw}\,}} (\mathcal {F})$ denotes the minimum treewidth of the graphs in $\mathcal {F}$. For the other case, where $\mathcal {F}$ contains a $P_3$-subgraph-free graph, we present a polynomial Turing kernelization. Our results extend to $\mathcal {F}$-Subgraph-Free Deletion.

B. M. P. Jansen: Supported by NWO Gravitation grant “Networks”.