Domination, Eternal Domination, and Clique Covering

TL;DR

This paper explores relationships between various graph parameters related to guarding and covering, providing characterizations for specific classes of graphs and inequalities among these parameters.

Contribution

It offers new characterizations of bipartite, triangle-free, and tree graphs where domination and eternal domination numbers coincide, and analyzes inequalities among key graph invariants.

Findings

Characterization of bipartite graphs with domination numbers equal to two

Characterization of triangle-free graphs with eternal domination number two

Identification of trees where m-eternal domination equals clique covering number

Abstract

Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique covering numbers of graph are explored in this paper. Among other results, we characterize bipartite and triangle-free graphs with domination and eternal domination numbers equal to two, trees with equal m-eternal domination and clique covering numbers, and two classes of graphs with equal domination, eternal domination and clique covering numbers.