Protecting a graph with mobile guards

Abstract

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.

Eternal domination on prisms of graphs

“…. Many open questions remain regarding eternal domination with respect to these bounding parameters, and in particular whether or not particular constructions force γ ∞ (G) to be equal to one of them (the survey [5] provides a nice overview of what is and is not known about eternal domination and its variants).…”

Eternally dominating large grids

“…For an overall picture and further references on the topic, we refer the reader to a recent survey on graph protection [21].…”

Disjoint dominating sets with a perfect matching

“…Various graph protection models have been considered in the literature, many of which are surveyed in [17]. In a graph protection problem, mobile guards aim to defend a graph from a sequence of attacks.…”