Inequalities of Nordhaus–Gaddum type for doubly connected domination number

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