Unique response Roman domination in graphs

Abstract

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 ∈ V ( G ) ∣ f ( v ) = i } for i = 0 , 1 , 2 , is a unique response Roman function if x ∈ V 0 implies | N ( x ) ∩ V 2 | ≤ 1 and x ∈ V 1 ∪ V 2 implies that | N ( x ) ∩ 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.