A note on k-Roman graphs

Let G = (V,E) be a graph and let k be a positive integer. A subset D of V (G) is a k-dominating set of G if every vertex in V (G) \D has at least k neighbours in D. The k-domination number Υk(G) is the minimum cardinality of a k-dominating set of 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. The weight of a Roman k-dominating function is the value [formula] and the minimum weight of a Roman k-dominating function on G is called the Roman k-domination number Υk(G) of G. A graph G is said to be a k-Roman graph if ΥkR(G) = 2Υk(G) . In this note we study k-Roman graphs.