Vol. 12, No. 7, 2019

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

Lauren Keough and David Shane

Vol. 12 (2019), No. 7, 1175–1181
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 n2 2 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.

Keywords
Nordhaus–Gaddum inequalities, dominating sets
Mathematical Subject Classification 2010
Primary: 05C35, 05C69
Milestones
Received: 5 December 2018
Revised: 18 March 2019
Accepted: 21 March 2019
Published: 12 October 2019

Communicated by Kenneth S. Berenhaut
Authors
Lauren Keough
Department of Mathematics
Grand Valley State University
Allendale, MI
United States
David Shane
Michigan State University
East Lansing, MI
United States