Inequalities of Nordhaus–Gaddum type for doubly connected domination number

Abstract

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 ${\gamma }_{cc}\left(G\right)$ is the minimum size of such a set. We prove that when $G$ and $\overline{G}$ are both connected of order $n$, ${\gamma }_{cc}\left(G\right)+{\gamma }_{cc}\left(\overline{G}\right)\le n+3$ and we describe the two infinite families of extremal graphs achieving the bound.