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 γ m ∞ G . In this paper, we find γ m ∞ P n , 1 and γ m ∞ P n , 3 for n ≡ 0 mod 4 . We also find upper bounds for γ m ∞ P n , 2 and γ m ∞ P n , 3 when n is arbitrary.
Highlights
The term graph protection refers to the process of placing guards or mobile agents in order to defend against a sequence of attacks on a network
The m-eternal domination numbers for cycles Cn and paths Pn were found by Goddard et al [3] as follows: γ∞ m ðCnÞ = dn/3e and γ∞ m ðPnÞ = dn/2e
A generalized Petersen graph Pðn, kÞ is a graph with vertex set V ∪ U = fv1, v2, ⋯, vng ∪ fu1, u2, ⋯, ung and edge set E = ∪fvivi+1, viu1, uiui+kg with vn+1 = v1, un+1 = u1, and 1 ≤ i ≤ n and 1 ≤ k ≤ bn/2c; see [6] for more information on generalized Petersen graph
Summary
The term graph protection refers to the process of placing guards or mobile agents in order to defend against a sequence of attacks on a network.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
