## Involve: A Journal of Mathematics

• Involve
• Volume 12, Number 7 (2019), 1175-1181.

### 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 $G¯$. In this spirit Wagner proved that any graph $G$ on $n$ vertices satisfies $∂(G)+∂(G¯)≥2n$, where $∂(G)$ is the number of dominating sets in a graph $G$. In the same paper he commented that proving an upper bound for $∂(G)+∂(G¯)$ among all graphs on $n$ vertices seems to be much more difficult. Here we prove an upper bound on $∂(G)+∂(G¯)$ and prove that any graph maximizing this sum has minimum degree at least $⌊n∕2⌋−2$ and maximum degree at most $⌈n∕2⌉+1$. We conjecture that the complete balanced bipartite graph maximizes $∂(G)+∂(G¯)$ and have verified this computationally for all graphs on at most $10$ vertices.

#### Article information

Source
Involve, Volume 12, Number 7 (2019), 1175-1181.

Dates
Revised: 18 March 2019
Accepted: 21 March 2019
First available in Project Euclid: 26 October 2019

https://projecteuclid.org/euclid.involve/1572055227

Digital Object Identifier
doi:10.2140/involve.2019.12.1175

Mathematical Reviews number (MathSciNet)
MR4023345

Zentralblatt MATH identifier
07140472

#### Citation

Keough, Lauren; Shane, David. Toward a Nordhaus–Gaddum inequality for the number of dominating sets. Involve 12 (2019), no. 7, 1175--1181. doi:10.2140/involve.2019.12.1175. https://projecteuclid.org/euclid.involve/1572055227

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