Toward a Nordhaus–Gaddum inequality for the number of dominating sets

Abstract

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex of $G$ is either in $S$ or is adjacent to a vertex in $S$. Nordhaus–Gaddum inequalities relate a graph $G$ to its complement $\overline{G}$. In this spirit Wagner proved that any graph $G$ on $n$ vertices satisfies $\partial \left(G\right)+\partial \left(\overline{G}\right)\ge 2^n$, where $\partial \left(G\right)$ is the number of dominating sets in a graph $G$. In the same paper he commented that proving an upper bound for $\partial \left(G\right)+\partial \left(\overline{G}\right)$ among all graphs on $n$ vertices seems to be much more difficult. Here we prove an upper bound on $\partial \left(G\right)+\partial \left(\overline{G}\right)$ and prove that any graph maximizing this sum has minimum degree at least $\lfloor n∕2\rfloor -2$ and maximum degree at most $\lceil n∕2\rceil +1$. We conjecture that the complete balanced bipartite graph maximizes $\partial \left(G\right)+\partial \left(\overline{G}\right)$ and have verified this computationally for all graphs on at most $10$ vertices.