Inequalities of Nordhaus-Gaddum type for doubly connected domination number

A set S of vertices of a connected graph G is a doubly connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraphs induced by S and V-S are connected. The doubly connected domination number@c"c"c(G) is the minimum size of such a set. We prove that when G and G@? are both connected of order n, @c"c"c(G)+@c"c"c(G@?)@?n+3 and we describe the two infinite families of extremal graphs achieving the bound.