On the Tractability of Defensive Alliance Problem

TL;DR

This paper investigates the computational complexity of the defensive alliance problem in graphs, showing polynomial-time solvability for degree at most 5, NP-completeness at degree 6, and providing fixed-parameter algorithms for certain graph parameters.

Contribution

It establishes the boundary between tractable and intractable cases based on maximum degree and offers FPT algorithms for parameters like twin cover and distance to clique.

Findings

Polynomial-time solvable for max degree ≤ 5

NP-Complete for max degree = 6

FPT algorithms for twin cover and distance to clique

Abstract

Given a graph $G = (V, E)$, a non-empty set $S \subseteq V$ is a defensive alliance, if for every vertex $v \in S$, the majority of its closed neighbours are in $S$, that is, $|N_G[v] \cap S| \geq |N_G[v] \setminus S|$. The decision version of the problem is known to be NP-Complete even when restricted to split and bipartite graphs. The problem is \textit{fixed-parameter tractable} for the parameters solution size, vertex cover number and neighbourhood diversity. For the parameters treewidth and feedback vertex set number, the problem is W[1]-hard. \\ \hspace*{2em} In this paper, we study the defensive alliance problem for graphs with bounded degree. We show that the problem is \textit{polynomial-time solvable} on graphs with maximum degree at most 5 and NP-Complete on graphs with maximum degree 6. This rules out the fixed-parameter tractability of the problem for the parameter maximum degree of the graph. We also consider the problem from the standpoint of parameterized complexity. We provide an FPT algorithm using the Integer Linear Programming approach for the parameter distance to clique. We also answer an open question posed in \cite{AG2} by providing an FPT algorithm for the parameter twin cover.