On Structural Parameterizations of the Offensive Alliance Problem

TL;DR

This paper investigates the computational complexity of finding minimal offensive alliances in graphs, revealing that the problem remains hard even under various structural restrictions of the input graph.

Contribution

It demonstrates that the Offensive Alliance problem is W[1]-hard when parameterized by several structural graph parameters, deepening understanding of its computational difficulty.

Findings

W[1]-hardness for feedback vertex set number

W[1]-hardness for treewidth

W[1]-hardness for pathwidth and treedepth

Abstract

The Offensive Alliance problem has been studied extensively during the last twenty years. A set $S\subseteq V$ of vertices is an offensive alliance in an undirected graph $G=(V,E)$ if each $v\in N(S)$ has at least as many neighbours in $S$ as it has neighbours (including itself) not in $S$. We study the parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. Our focus here lies on parameters that measure the structural properties of the input instance. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph.