weak roman domination stable graphs upon edge addition

A roman dominating function (rdf) on a graph g is a function f : v (g) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. a vertex u with f(u) = 0 is said to be 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 0 : v (g) → {0, 1, 2} defined by f 0 (u) = 1, f 0 (v) = f(v) − 1 and f 0 (w) = f(w) if w ∈ v − {u, v}, has no undefended vertex. a graph g is said to be roman domination stable upon edge addition, or just γr-ea-stable, if γr(g + e) = γr(g) for any edge e /∈ e(g). we extend this concept to a weak roman dominating function as follows: a graph g is said to be weak roman domination stable upon edge addition, or just γr-ea-stable, if γr(g + e) = γr(g) for any edge e /∈ e(g). in this paper, we study γr-ea-stable graphs, obtain bounds for γr-ea-stable graphs and characterize γr-ea-stable trees which attain the bound.