Effect of vertex deletion on the weak Roman domination number of a graph

Let G = (V, E) be a graph and f : V -> {0, 1, 2} be a function. The weight of a vertex u is an element of V is f (u) and a vertex u with weight f (u) = 0 is said to be undefended with respect to f , if it is not adjacent to a vertex with positive weight. The function f 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 -> {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 weight of f is w(f) = Sigma(v is an element of V) f (v). The weak Roman domination number, denoted by gamma(r) (G), is the minimum weight of a weak Roman dominating function on G. In this paper we examine the effects on gamma(r) (G) when G is modified by deleting a vertex. (C) 2017 Kalasalingam University. Production and Hosting by Elsevier B.V.