Abstract
In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. We consider the “all guards move” of the eternal dominating set problem. In which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current position after each attack. If the new formed set of guards is still a dominating set of the graph then we successfully defended the attack. Our goal is to find the minimum number of guards required to eternally protect the graph. We call this number the m-eternal domination number and we denote it by . In this paper we find the eternal domination number of Jahangir graph Js,m for s=2,3 and arbitrary m. We also find the domination number for J3,m .
Highlights
In graph protection, mobile agents or guards are placed on vertices in order to defend against a sequence of attacks on a network
In which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current position after each attack
We call this number the m-eternal domination number and we denote it by γ
Summary
Mobile agents or guards are placed on vertices in order to defend against a sequence of attacks on a network. In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. Was defined by Goldwasser et al [7] as follows: Let G be a graph with a dominating set of cardinality k.
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