Involve: A Journal of Mathematics
- Volume 12, Number 7 (2019), 1175-1181.
Toward a Nordhaus–Gaddum inequality for the number of dominating sets
A dominating set in a graph is a set of vertices such that every vertex of is either in or is adjacent to a vertex in . Nordhaus–Gaddum inequalities relate a graph to its complement . In this spirit Wagner proved that any graph on vertices satisfies , where is the number of dominating sets in a graph . In the same paper he commented that proving an upper bound for among all graphs on vertices seems to be much more difficult. Here we prove an upper bound on and prove that any graph maximizing this sum has minimum degree at least and maximum degree at most . We conjecture that the complete balanced bipartite graph maximizes and have verified this computationally for all graphs on at most vertices.
Involve, Volume 12, Number 7 (2019), 1175-1181.
Received: 5 December 2018
Revised: 18 March 2019
Accepted: 21 March 2019
First available in Project Euclid: 26 October 2019
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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