Parameterized Complexity of Locally Minimal Defensive Alliances

TL;DR

This paper investigates the computational complexity of finding maximum locally minimal defensive alliances in graphs, revealing NP-completeness, fixed-parameter tractability under certain parameters, and polynomial-time solvability for bounded treewidth graphs.

Contribution

It establishes the NP-completeness of the problem, provides a randomized FPT algorithm, and characterizes its complexity with respect to various graph parameters.

Findings

NP-complete even for planar graphs

FPT algorithm for exact connected case by solution size

Polynomial-time for graphs with bounded treewidth

Abstract

A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours is in $S$. We consider the notion of local minimality in this paper. We are interested in locally minimal defensive alliance of maximum size. We also look at connected version of defensive alliance. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) Locally Minimal Defensive Alliance is NP-complete, even when restricted to planar graphs, (2) a randomized FPT algorithm for Exact Connected Locally Minimal Defensive Alliance parameterized by solution size, (3) Locally Minimal Defensive Alliance is fixed-parameter tractable (FPT) when parametrized by neighbourhood diversity, (4) Locally Minimal Defensive Alliance parameterized by treewidth is W[1]-hard and thus not FPT (unless FPT=W[1]), (5) Locally Minimal Defensive Alliance can be solved in polynomial time for graphs of bounded treewidth.