Roman k-domination in graphs

Abstract

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f: V (G) → {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices v1, v2,..., vk with f(vi) = 2 for i = 1, 2,..., k. Phe weight of a Roman k-dominating function is the value f(V (G)) =u∈V (G) f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number γkR(G) of G.Note that the Roman 1-domination number γ1R(G) is the usual Roman domination number γR(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number. © 2009 The Korean Mathematical Society.