(Sub)linear Kernels for Edge Modification Problems Toward Structured Graph Classes

Abstract

In a (parameterized) graph edge modification problem, we are given a graph G, an integer k and a (usually well-structured) class $\mathcal {G}$ of graphs, and asked whether it is possible to transform G into a graph $G' \in \mathcal {G}$ by adding and/or removing at most k edges. Parameterized graph edge modification problems received considerable attention in the last decades. In this paper, we focus on finding small kernels for edge modification problems. One of the most studied problems is the Cluster Editing problem, in which the goal is to partition the vertex set into a disjoint union of cliques. Even if a 2k-vertex kernel exists for Cluster Editing, this kernel does not reduce the size of the instance in most cases. Therefore, we explore the question of whether linear kernels are a theoretical limit in edge modification problems, in particular when the target graph class is very structured (such as a partition into cliques for instance). We prove, as far as we know, the first sublinear kernel for an edge modification problem. Namely, we show that Clique + Independent Set Deletion, which is a restriction of Cluster Deletion, admits a kernel of size $O(k/\log k)$. We also obtain small kernels for several other edge modification problems. We first show that Cluster Deletion admits a 2k-vertex kernel as Cluster Editing, improving the previous 4k-vertex kernel. We prove that (Pseudo-)Split Completion (and the equivalent (Pseudo-)Split Deletion) admits a linear kernel, improving the existing quadratic kernel. We also prove that Trivially Perfect Completion admits a quadratic kernel (improving the cubic kernel), and finally prove that its triangle-free version (Starforest Deletion) admits a linear kernel, which is optimal under the Exponential Time Hypothesis.