A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees

TL;DR

This paper introduces the first non-trivial polynomial kernel of size O(k^5) for the deletion problem that transforms a graph into a scattered class of cliques and trees, advancing kernelization in graph modification problems.

Contribution

It provides the first polynomial kernel for deletion to scattered classes of cliques and trees, a significant step in kernelization for graph deletion problems.

Findings

Developed a kernel with O(k^5) vertices for the problem.

Established the first non-trivial polynomial kernel for this class of problems.

Advances understanding of kernelization complexity in graph deletion problems.

Abstract

The class of graph deletion problems has been extensively studied in theoretical computer science, particularly in the field of parameterized complexity. Recently, a new notion of graph deletion problems was introduced, called deletion to scattered graph classes, where after deletion, each connected component of the graph should belong to at least one of the given graph classes. While fixed-parameter algorithms were given for a wide variety of problems, little progress has been made on the kernelization complexity of any of them. In this paper, we present the first non-trivial polynomial kernel for one such deletion problem, where, after deletion, each connected component should be a clique or a tree - that is, as densest as possible or as sparsest as possible (while being connected). We develop a kernel consisting of O(k^5) vertices for this problem.