Weak Roman Bondage Number of a Graph

A Roman dominating function (RDF) on a graph G is a labelling f : V (G) -> {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. A vertex u with f (u) = 0 is said to be undefended with respect to f if it is not adjacent to a vertex v with the positive weight. A function f : V (G) -> {0, 1, 2} is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f' : V (G) -> {0, 1, 2} defined by f'(u) = 1, f' (v) = f(v) - 1 and f'(w) = f(w) if w is an element of V - {u, v}, has no undefended vertex. The Roman bondage number bR(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E' subset of E(G) for which gamma(R)(G - E') > gamma(R)(G). We extend this concept to a weak Roman dominating function as follows: The weak Roman bondage number b(r)(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E' subset of E(G) for which gamma(r)(G - E') > gamma(r)(G). In this paper we determine the exact values of the weak Roman bondage number for paths, cycles and complete bipartite graphs. We obtain bounds for trees and unicyclic graphs and characterize the extremal graphs.