Involve: A Journal of Mathematics

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

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

Lauren Keough and David Shane

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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 n22 and maximum degree at most n2+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
Received: 5 December 2018
Revised: 18 March 2019
Accepted: 21 March 2019
First available in Project Euclid: 26 October 2019

Permanent link to this document
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

Subjects
Primary: 05C35: Extremal problems [See also 90C35] 05C69: Dominating sets, independent sets, cliques

Keywords
Nordhaus–Gaddum inequalities dominating sets

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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