A note on Roman domination in graphs

Abstract

Let G = ( V , E ) be a simple graph. A subset S ⊆ V is a dominating set of G, if for any vertex u ∈ V - S , there exists a vertex v ∈ S such that uv ∈ E . The domination number of G, γ ( G ) , equals the minimum cardinality of a dominating set. A Roman dominating function on graph G = ( V , E ) is a function f : V → { 0 , 1 , 2 } satisfying the condition that every vertex v for which f ( v ) = 0 is adjacent to at least one vertex u for which f ( u ) = 2 . The weight of a Roman dominating function is the value f ( V ) = ∑ v ∈ V f ( v ) . The Roman domination number of a graph G, denoted by γ R ( G ) , equals the minimum weight of a Roman dominating function on G. In this paper, for any integer k ( 2 ⩽ k ⩽ γ ( G ) ) , we give a characterization of graphs for which γ R ( G ) = γ ( G ) + k , which settles an open problem in [E.J. Cockayne, P.M. Dreyer Jr, S.M. Hedetniemi et al. On Roman domination in graphs, Discrete Math. 278 (2004) 11–22].