On the Unique Response Roman Domination Numbers of Graphs

Abstract

Let G be a graph with vertex set V(G). A function f: V(G) → {0, 1, 2} with the ordered partition (V0, V1, V2) of V(G), where Vi = {V∈V(G) | f(V) = i} for i = 0, 1, 2, is a Roman dominating function if x ∈ V0 implies |N(x)∩V2|≥ 1. It is a unique response Roman function if x ∈ V0 implies |N(x) ≥ V2|≤ 1 and x ∈ V1 ∪ V2 implies that |N(x) ∩ V2| = 0. A function f: V(G) → {0, 1, 2} is a unique response Roman dominating function if it is both a unique response Roman function and a Roman dominating function. The unique response Roman domination number, denoted by uR(G), of G is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination of graphs, and provide some graphs whose unique response Roman domination number equals to the independent Roman domination number.