2-domination subdivision number of graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a 2-dominating set of G if S dominates every vertex of V (G) \ S at least twice. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set of G. The 2-domination subdivision number sdγ2(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2domination number. In this paper, we establish upper bounds on the 2-domination subdivision number for arbitrary graphs in terms of vertex degree and for several graph classes. Then we present some conditions on G which are sufficient to imply that sdγ2(G) = 1.