Weakly connected Roman domination in graphs

Abstract

A Roman dominating function on a graph G=(V,E) is defined to be a function f:V→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. A dominating set D⊆V is a weakly connected dominating set of G if the graph (V,E∩(D×V)) is connected. We define a weakly connected Roman dominating function on a graph G to be a Roman dominating function such that the set {u∈V:f(u)∈{1,2}} is a weakly connected dominating set of G. The weight of a weakly connected Roman dominating function is the value f(V)=∑u∈Vf(u). The minimum weight of a weakly connected Roman dominating function on a graph G is called the weakly connected Roman domination number of G and is denoted by γRwc(G). In this paper, we initiate the study of this parameter.