Weak Roman subdivision number of graphs

A Roman dominating function (RDF) on a graph G is a function f : V(G) -> {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 vertex u with f(u) = 0 is undefended if it is not adjacent to a vertex with f(v) > 0. The 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 domination subdivision number of a graph G is the minimum number of edges that. must be subdivided in order to increase the Roman domination number of G. We introduce the concept of weak Roman subdivision number of a graph G, denoted by sd(gamma r) (G) as the minimum number of edges that must be subdivided in order to increase the weak Roman domination number of G. in this paper, we determine the exact values of the weak Roman subdivision number for paths, cycles and complete bipartite graphs. We obtain bounds for the weak Roman subdivision number of a graph and characterize the extremal graphs.