Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1

Let G = (V, E) be a simple graph with vertex set V and edge set E, and let f be a function f : V 7 -> {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 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 fu : V 7 -> {0, 1, 2}, defined by fu(u) = 1, fu(v) = f (v) - 1 and fu(w) = f(w) if w is an element of V - {u, v}, has no undefended vertex. The weight of f is w(f) = Zv is an element of V f (v). The weak Roman domination number, denoted gamma r(G), is the minimum weight of a WRDF in G. The domination number, denoted gamma(G), is the minimum cardinality of a dominating set in G. In this paper, we give some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 (gamma r(T) = gamma(T) + 1) by recursion and construction.

Automated summary

In this paper, some sufficient conditions are given for a tree to have its weak Roman domination number equal to its domination number plus 1 (gamma r(T) = gamma(T) + 1) by recursion and construction.