Preprocessing Subgraph and Minor Problems: When Does a Small Vertex Cover Help?

TL;DR

This paper investigates kernelization for problems parameterized by vertex cover size, providing conditions for polynomial kernels, new kernel results for minor-free deletion, and contrasting complexities of induced versus minor containment problems.

Contribution

It offers a unified characterization for when problems admit polynomial kernels based on vertex cover, and presents new kernelization results for F-Minor-Free Deletion and other problems.

Findings

Polynomial kernels exist for F-Minor-Free Deletion and similar problems.

Testing for a complete graph as a minor admits a polynomial kernel.

Testing for an induced path as a minor likely does not admit a polynomial kernel.

Abstract

We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Transversal, Chordal Deletion, Eta Transversal, or Long Path, parameterized by the size of a vertex cover, but also imply new polynomial kernels for problems like F-Minor-Free Deletion, which is to delete at most k vertices to obtain a graph with no minor from a fixed finite set F. While our characterization captures many interesting problems, the kernelization complexity landscape of parameterizations by vertex cover is much more involved. We demonstrate this by several results about induced subgraph and minor containment testing, which we find surprising. While it was known that testing for an induced complete subgraph has no polynomial kernel unless NP is in coNP/poly, we show that the problem of testing if a graph contains a complete graph on t vertices as a minor admits a polynomial kernel. On the other hand, it was known that testing for a path on t vertices as a minor admits a polynomial kernel, but we show that testing for containment of an induced path on t vertices is unlikely to admit a polynomial kernel.