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 inequailties relate a graph $G$ to its complement $\bar{G}$. In this spirit Wagner proved that any graph $G$ on $n$ vertices satisfies $\partial(G)+\partial(\bar{G})\geq 2^n$ where $\partial(G)$ is the number of dominating sets in a graph $G$. In the same paper he comments that an upper bound for $\partial(G)+\partial(\bar{G})$ among all graphs on $n$ vertices seems to be much more difficult. Here we prove an upper bound on $\partial(G)+\partial(\bar{G})$ and prove that any graph maximizing this sum has minimum degree at least $\lfloor n/2\rfloor-2$ and maximum degree at most $\lfloor n/2\rfloor+1$. We conjecture that the complete balanced bipartite graph maximizes $\partial(G)+\partial(\bar{G})$ and have verified this computationally for all graphs on at most $10$ vertices.