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. The weight of a Roman k-dominating function is the value f(V (G)) = u2V (G) f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number ∞kR(G) of G. Note that the Roman 1-domination number ∞1R(G) is the usual Roman domination number ∞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. 1. Terminology and introduction We consider finite, undirected and simple graphs G with vertex set V (G) and edge set E(G). The number of vertices |V (G)| of a graph G is called the order of G and is denoted by n = n(G). The open neighborhood N(v) = NG(v) of a vertex v consists of the vertices adjacent to v and d(v) = dG(v) = |N(v)| is the degree of v. The closed neigh- borhood of a vertex v is defined by N(v) = NG(v) = N(v)({v}. The maximum degree of a graph G is denoted by ¢(G) = ¢. For a subset S µ V (G), we define N(S) = NG(S) = S v2S N(v), N(S) = NG(S) = N(S) ( S, and G(S) is the subgraph induced by S. The complement of a graph G is denoted by G. If !(G) is the number of components of G and m(G) = |E(G)|, then c(G) = m(G) i n(G) + !(G) is the well-known cyclomatic number of G. A graph is a cactus graph if all its cycles are edge-disjoint. We write Kn for the complete graph of order n, and Kp,q for the complete bipartite graph with bipartition X,Y such that |X| = p and |Y | = q.