Upper bounds on Roman domination numbers of graphs

A Roman dominating function of a graph G is a function f: V (G) -> {0. 1.2} such that whenever f (v) = 0 there exists a vertex u adjacent to v with f (u) = 2. The weight of f is w(f) = Sigma(v epsilon V(G))f(v). The Roman domination number gamma(R)(G) of G is the minimum weight of a Roman dominating function of G. This paper establishes a sharp upper bound on the Roman domination numbers of graphs with minimum degree at least 3. An upper bound on the Roman domination numbers of connected, big-claw-free and big-net-free graphs is also given. (C) 2011 Elsevier B.V. All rights reserved.