On the parameterized complexity of clique elimination distance

Bulian and Dawar [Algorithmica, 2016] introduced the notion of elimination distance in an effort to define new tractable parameterizations for graph problems and showed that deciding whether a given graph has elimination distance at most k to any minor-closed class of graphs is fixed-parameter tractable parameterized by k [Algorithmica, 2017]. In this paper, we consider the problem of computing the elimination distance of a given graph to the class of cluster graphs and initiate the study of the parameterized complexity of a more general version – that of obtaining a modulator to such graphs. That is, we study the (η, Clq)-Elimination Deletion problem ((η, Clq)-ED Deletion) where, for a fixed η, one is given a graph G and k ∈ N and the objective is to determine whether there is a set S ⊆ V (G) such that the graph G − S has elimination distance at most η to the class of cluster graphs. Our main result is a polynomial kernelization (parameterized by k) for this problem. As components in the proof of our main result, we develop a k^O(ηk+^η2)n^O(1)-time fixed-parameter algorithm for (η, Clq)-ED Deletion and a polynomial-time factor-min{O(η · opt · log² n), opt^O(1)} approximation algorithm for the same problem.