Total outer-Connected domination subdivision numbers in graphs

A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph G[V\S] induced by V\S is connected. The total outer-connected domination numberγtoc(G) is the minimum size of such a set. The total outer-connected domination subdivision numbersdγtoc(G) is the minimum number of edges that must be subdivided in order to increase the total outer-connected domination number. We prove the existence of sdγtoc(G) for every connected graph G of order at least 3 and give upper bounds on it in some classes of graphs.