Computing m-Eternal Domination Number of Cactus Graphs in Linear Time

TL;DR

This paper presents a linear-time algorithm to compute the m-eternal domination number of cactus graphs, considering two variants, advancing understanding of guard placement strategies in graph defense scenarios.

Contribution

It introduces a linear-time algorithm for calculating the m-eternal domination number of cactus graphs for two variants, with new tools for bounds and analysis.

Findings

Algorithm computes minimum guards in linear time

Handles two variants of guard placement

Provides bounds and analysis tools for cactus graphs

Abstract

In m-eternal domination attacker and defender play on a graph. Initially, the defender places guards on vertices. In each round, the attacker chooses a vertex to attack. Then, the defender can move each guard to a neighboring vertex and must move a guard to the attacked vertex. The m-eternal domination number is the minimum number of guards such that the graph can be defended indefinitely. In this paper, we study the m-eternal domination number of cactus graphs. We consider two variants of the m-eternal domination number: one allows multiple guards to occupy a single vertex, the second variant requires the guards to occupy distinct vertices. We develop several tools for obtaining lower and upper bounds on these problems and we use them to obtain an algorithm which computes the minimum number of required guards of cactus graphs for both variants of the problem.