On the Roman bondage Number of a Graph

A Roman dominating function on a graph G = (V,E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number bR(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E′ ⊆ E(G) for which γR(G−E′) > γR(G). In this note we first present sharp bounds for bR(G) and then we initiate the study of the Roman k-bondage number in graphs. Some of our results extend those given by Jafari Rad and Volkmann in 2011 for the Roman bondage number.