Parameterized Complexity of Upper Edge Domination

TL;DR

This paper investigates the parameterized complexity of the Upper Edge Dominating Set problem, providing kernelization results and an FPT algorithm, advancing understanding of its computational tractability.

Contribution

It introduces a polynomial kernel and an FPT algorithm for the Upper EDS problem, a maximization variant of the classical Edge Dominating Set.

Findings

Kernel with at most 4k^2 - 2 vertices

FPT algorithm with runtime 2^{O(k)} * n^{O(1)}

Enhanced understanding of the problem's fixed-parameter tractability

Abstract

In this paper we study a maximization version of the classical Edge Dominating Set (EDS) problem, namely, the Upper EDS problem, in the realm of Parameterized Complexity. In this problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a minimal edge dominating set of size at least $k$. We obtain the following results for Upper EDS. We prove that Upper EDS admits a kernel with at most $4k^2-2$ vertices. We also design a fixed-parameter tractable (FPT) algorithm for Upper EDS running in time $2^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(1)}$.