Rainbow domination numbers of generalized Petersen graphs

Abstract

Abstract Domination and its variations in graphs are natural models for the location problems in operations research. In this paper, we investigate the rainbow domination number of graphs, which was introduced by Bresar, Henning and Rall. Given a graph G and a positive integer t, a t-rainbow dominating function of G is a function f from vertex set to the set of all subsets of {1, 2, ⋅⋅⋅, t} such that for any vertex v with f ( v ) = ϕ , we have ⋃ u ∈ N ( v ) f ( u ) = { 1 , 2 , ⋯ , t } . The t-rainbow domination problem is to determine the t-rainbow domination number γrt(G) of G, that is the minimum value of ∑v ∈ V(G)|f(v)|, where f runs over all t-rainbow dominating functions of G. The domination number and its variations of generalized Petersen graphs P(n, k) are widely investigated. The exact values of γr2(P(n, 1)) and γr3(P(n, 1)) are already determined in [11, 12]. In this paper, we determine the exact values of γrt(P(n, 1)) for any t ≥ 8 and t = 4 and prove that γ r t ( P ( 2 k , k ) ) = 4 k for t ≥ 6, where P(2k, k) is a special graph.