Domination and Eternal Domination of Jahangir Graph

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 () m G γ ∞. In this paper we find the eternal domination number of Jahangir graph , s m J for 2,3 s = and arbitrary m. We also find the domination number for 3,m J .

Eternal Domination of Generalized Petersen Graph

“…A generalized Jahangir graph J s,m for m ≥ 2 is a graph on sm + 1 vertices, i.e., a graph consisting of a cycle C sm with one additional vertex which is adjacent to m vertices of C sm at distance s from each other on C sm ; see [8] for more information on the Jahangir graph. In [9], we found γ ∞ m ðJ s,m Þ for s ≡ 2, 3. In [10], we found γ ∞ m ðJ s,m Þ for s, m are arbitraries.…”

“…In [9], we found γ ∞ m ðJ s,m Þ for s ≡ 2, 3. In [10], we found γ ∞ m ðJ s,m Þ for s, m are arbitraries.…”