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∈V is f(u)and a vertex u with weightf(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∈V−{u,v}, has no undefended vertex. The weight of f is w(f)=∑v∈Vf(v). The weak Roman domination number, denoted by γr(G), is the minimum weight of a weak Roman dominating function on G. In this paper we examine the effects on γr(G)when G is modified by deleting a vertex. Keywords: Weak roman domination number, Changing and unchanging, Vertex deletion

Abstract licence: CC BY-NC-ND

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