| Motivated by an article by lan Stewart[4](Denfend the Roman Empire!, scientific American, Dec. 1999, pp. 136-138), M.A. Henning and S.T. Hedetniemi [1] explore a new strategy of defending the Roman Empire that has the potential of saving the Emperor Constantine the Great substantial costs of maintaining legions, while still defending the Roman Empire. In graph theoretic terminology, let G = (V, E) be a graph and let f be function f : V (?) {0,1,2}. A vertex u with f(u) = 0 is said to be undefended with respect to f if it is not adjacent to a vertex with positive weight.the function f is a Weak Roman domination function (WRDF) if each vertex 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, denoteγr(G) is the minimum weight of a WRDF in G. In this paper, suppose T is a simple connected graph obtained from T1 and T2 by adding a new edge joining v1∈V(T1) and v2∈V(T2), we discuss the relationship betweenγr(T) andγr(T1) +γr(T 2), and we given necessary and sufficient condition ofγr(T) =γr(T1) +γr(T2) -1. At the same time, if G is a simple connected graph with the connectivity is 1, we also discussed impact on its weak Roman domination number if we remove one of its cut-point.
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