ROMAN k-DOMINATION IN GRAPHS

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 v(1), v(2), ..., v(k) with f(v(i)) = 2 for i = 1, 2,..., k. The weight of a Roman k-dominating function is the value f(V(G)) = Sigma(u is an element of V(G)) f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number gamma(kR)(G) of G. Note that the Roman I-domination number gamma(1R)(G) is the usual Roman domination number gamma(R)(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.