# On the m-eternal Domination Number of Cactus Graphs

## TL;DR

This paper investigates the m-eternal domination number of cactus graphs, providing new bounds, proving equality of variants for Christmas cactus graphs, and offering a linear-time computation algorithm.

## Contribution

It introduces a new upper bound for cactus graphs, proves the equality of three variants for Christmas cactus graphs, and develops a linear-time algorithm for their computation.

## Key findings

- New upper bound for m-eternal domination number of cactus graphs.
- Equality of three domination variants for Christmas cactus graphs.
- Linear-time algorithm for computing the m-eternal domination number.

## Abstract

Given a graph $G$, guards are placed on vertices of $G$. Then vertices are subject to an infinite sequence of attacks so that each attack must be defended by a guard moving from a neighboring 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, that is, connected graphs where each edge lies in at most two cycles, and we consider three variants of the m-eternal domination number: first variant allows multiple guards to occupy a single vertex, second variant does not allow it, and in the third variant additional "eviction" attacks must be defended. We provide a new upper bound for the m-eternal domination number of cactus graphs, and for a subclass of cactus graphs called Christmas cactus graphs, where each vertex lies in at most two cycles, we prove that these three numbers are equal. Moreover, we present a linear-time algorithm for computing them.