Towards optimal kernel for connected vertex cover in planar graphs

TL;DR

This paper improves the kernel size for the connected vertex cover problem in planar graphs from 4k to approximately 3.67k, advancing understanding of kernelization limits in this setting.

Contribution

The paper presents a new kernelization result with an 11/3 k vertex bound for the connected vertex cover problem in planar graphs, improving previous bounds.

Findings

Established a (11/3)k-vertex kernel for the problem

Demonstrated that the previous 4k kernel is not optimal

Motivated further research into kernelization bounds

Abstract

We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP is a subset of coNP/poly), for planar graphs Guo and Niedermeier [ICALP'08] showed a kernel with at most 14k vertices, subsequently improved by Wang et al. [MFCS'11] to 4k. The constant 4 here is so small that a natural question arises: could it be already an optimal value for this problem? In this paper we answer this quesion in negative: we show a (11/3)k-vertex kernel for Connected Vertex Cover in planar graphs. We believe that this result will motivate further study in search for an optimal kernel.