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.