Inequalities of Nordhausgaddum type for doubly connected domination

Abstract

Many problems in extremal graph theory seek the extreme values of graph parameters on families of graphs. Results of Nordhaus-Gaddum type study the extreme values of the sum (or product) of a parameter on a graph and its complement, following the classic paper of Nordhaus and Gaddum solving these problems for the chromatic number on n- vertex graphs. In this paper, we study such problems for the 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 cc(G) is the minimum size of such a set. We prove that when G and G are both connected of order n, cc(G) +

cc(G) n + 2 and we describe the two in nite families of extremal graphs achieving the bound.