Unique response Roman domination in graphs
A function f:V(G)?{0,1,2} f : V ( G ) ? { 0 , 1 , 2 } is a Roman dominating function  if every vertex u u for which f(u)=0 f ( u ) = 0 is adjacent to at least one vertex v v for which f(v)=2 f ( v ) = 2 . A function f:V(G)?{0,1,2} f : V ( G ) ? { 0 , 1 , 2 } with the ordered partition (V0,V1,V2) ( V 0 , V 1 , V 2 ) of V(G) V ( G ) , where Vi={v?V(G)?f(v)=i} V i = { v ? V ( G ) ? f ( v ) = i } for i=0,1,2 i = 0 , 1 , 2 , is a unique response Roman function if x?V0 x ? V 0 implies |N(x)?V2|?1 | N ( x ) ? V 2 | ? 1 and x?V1?V2 x ? V 1 ? V 2 implies that |N(x)?V2|=0 | N ( x ) ? V 2 | = 0 . A function f:V(G)?{0,1,2} 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 G , denoted by uR(G) 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.
roman domination domination