Strong weakly connected domination subdivisible graphs

The weakly connected domination subdivision number sdγw(G) of a connected graph G is the minimum number of edges which must be subdivided (where each edge can be subdivided at most once) in order to increase the weakly connected domination number. The graph is strongγw-subdivisible if for each edge uv ∈ E(G) we have γw(Guv) > γw(G), where Guv is a graph G with subdivided edge uv. The graph is strong-γwk-subdivisible if sdγw(G) = k and subdivision of each of the k edges in G changes the weakly connected domination number of the graph obtained by these subdivisions. We constructively characterize all strong-γw-1subdivisible and strong-γw-2-subdivisible trees and give some properties of strong-γw-k-subdivisible graphs for k = 1, 2.