Inverse domination: stability, unique minimum dominating sets, and dual domination

Edwards, Timothy; Prier, David

doi:10.2140/involve.2025.18.885

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Inverse domination: stability, unique minimum dominating sets, and dual domination

Timothy Edwards and David Prier

Vol. 18 (2025), No. 5, 885–896

DOI: 10.2140/involve.2025.18.885

Abstract

An open question in graph theory asks whether the inverse domination number of an isolate-free graph is always less than or equal to the independence number of said graph. We discuss several classes of graphs for which this conjecture has already been proved and state general upper bounds on the inverse domination number. We also provide examples to demonstrate the complexity of this problem. This problem appears to be easier to address for graphs with a unique minimum dominating set or with maximum dual domination number. We end with two sections discussing these properties and their relation to this problem.

Keywords

graph theory, domination, inverse domination, stability, dual domination

Mathematical Subject Classification

Primary: 05C69

Milestones

Received: 11 March 2024

Accepted: 13 June 2024

Published: 13 November 2025

Communicated by John C. Wierman

Authors

Timothy Edwards
Department of Mathematics Gannon University Erie, PA United States

David Prier
Department of Mathematics Gannon University Erie, PA United States

Vol. 18, No. 5, 2025