Abstract
For a fixed finite family of graphs F, the F-Minor-Free Deletion problem takes as input a graph G and integer ℓ and asks whether a size-ℓ vertex set X exists such that G−X is F-minor-free. {K2}-Minor-Free Deletion and {K3}-Minor-Free Deletion encode 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. We show {P3}-Minor-Free Deletion parameterized by the feedback vertex number is MK[2]-hard. This rules out the existence of a polynomial kernel assuming NP⊈coNP/poly. Our hardness result generalizes to any F containing only graphs with a connected component of at least 3 vertices, using as parameter the vertex-deletion distance to treewidth mintw(F), where mintw(F) denotes the minimum treewidth of the graphs in F. For all other families F we present a polynomial Turing kernelization. Our results extend to F-Subgraph-Free Deletion.
Highlights
Motivated by the fact that Vertex Cover and Feedback Vertex Set, arguably the simplest F -Minor-Free Deletion problems, admit polynomial kernels when parameterized by the feedback vertex number, we set out to resolve the following question: Do all F -Minor-Free Deletion problems admit a polynomial kernel when parameterized by the feedback vertex number?
F Minor-Free Deletion and F -Subgraph-Free Deletion admit polynomial Turing kernels when parameterized by the vertex-deletion distance to a graph of treewidth min tw(F )
In this paper we showed that when F contains a forest and each graph in F has a connected component of at least three vertices, the F -Minor-Free Deletion and F -Subgraph-Free Deletion problems do not admit such a polynomial kernel unless NP ⊆ coNP/poly
Summary
One formal way to capture nontriviality of a graph problem is to measure how many vertex-deletions are needed to reduce the input graph to a graph class in which the problem can be solved in polynomial time. Problems can be solved in polynomial time on trees and forests, the structural graph parameter feedback vertex number (the minimum number of vertex deletions needed to make the graph acyclic, i.e., a forest) is a relevant measure of the distance of the input to a trivially solvable one. Previous research has shown that for the Vertex Cover problem, there is a polynomial kernel parameterized by the feedback vertex number [12] This preprocessing algorithm guarantees that inputs which are large with respect to their feedback vertex number can be efficiently reduced. Motivated by the fact that Vertex Cover and Feedback Vertex Set, arguably the simplest F -Minor-Free Deletion problems, admit polynomial kernels when parameterized by the feedback vertex number, we set out to resolve the following question: Do all F -Minor-Free Deletion problems admit a polynomial kernel when parameterized by the feedback vertex number?
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