On Eternal Domination of Generalized Js,m

Abstract

An eternal dominating set of a graph $G$ is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks, an attack is defended by moving one guard along an edge from its position to the attacked vertex. 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 positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the $m$-eternal domination model. The size of the smallest $m$-eternal dominating set is called the $m$-eternal domination number and is denoted by ${\gamma }_m^{\infty }\left(G\right)$. In this paper, we find the domination number of Jahangir graph $J_{s,m}$ for $s\equiv 1,2\left(\mathrm{mod}3\right)$, and the $m$-eternal domination numbers of $J_{s,m}$ for $s,m$ are arbitraries.