Defensive Alliances in Graphs

TL;DR

This paper investigates the computational complexity of finding minimal defensive alliances in graphs, revealing hardness results for various structural parameters and establishing NP-completeness on circle graphs.

Contribution

It proves the W[1]-hardness of the problem parameterized by several graph parameters and shows the non-existence of polynomial compression and subexponential algorithms under standard complexity assumptions.

Findings

W[1]-hardness for feedback vertex set, treewidth, pathwidth, and treedepth

No polynomial compression unless coNP ⊆ NP/poly

NP-complete on circle graphs

Abstract

A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours are in $S$. We study the parameterized complexity of the Defensive Alliance problem, where the aim is to find a minimum size defensive alliance. Our main results are the following: (1) The Defensive Alliance problem has been studied extensively during the last twenty years, but the question whether it is FPT when parameterized by feedback vertex set has still remained open. We prove 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; (2) the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP $\subseteq$ NP/poly, (3) it does not admit $2^{o(n)}$ algorithm under ETH, and (4) the Defensive Alliance problem on circle graphs is NP-complete.