On the meternal Domination Number of Cactus Graphs
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 meternal domination number is the minimum number of guards such that the graph can be defended indefinitely. In this paper we study the meternal 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 meternal 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 meternal 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 lineartime algorithm for computing them.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 DOI:
 10.48550/arXiv.1907.07910
 arXiv:
 arXiv:1907.07910
 Bibcode:
 2019arXiv190707910B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics