Unique response Roman domination in graphs

A function f : V(G) -> {0, 1, 2} is a Roman dominating function if every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. A function f : v(G) -> {0, 1, 2} with the ordered partition (V-0, V-1. V-2) of V(G), where V-i ={v is an element of V(G) | f(v) = i} for i = 0, 1, 2, is a unique response Roman function if X E V0 implies |N(x) boolean AND V-2| <= 1 and x is an element of V-1 boolean OR V-2 implies that |N(x) boolean AND V-2| = 0. A function f : V (G) -> {0, 1, 2} is a unique response Roman dominating function if it is a unique response Roman function and a Roman dominating function. The unique response Roman domination number of G, denoted by u(R)(G), is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination number of graphs and present bounds for this parameter. (C) 2011 Elsevier B.V. All rights reserved.