Connected domination in plane triangulations

Bryant, Felicity; Pavelescu, Elena

doi:10.2140/involve.2025.18.555

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Connected domination in plane triangulations

Felicity Bryant and Elena Pavelescu

Vol. 18 (2025), No. 3, 555–566

DOI: 10.2140/involve.2025.18.555

Abstract

A set of vertices of a graph $G$ with the property that each vertex of $G$ is either in the set or is adjacent to a vertex in the set is called a dominating set of $G$ . If, additionally, the set of vertices induces a connected subgraph of $G$ then the set is a connected dominating set of $G$ . The domination number $γ (G)$ of $G$ is the smallest number of vertices in a dominating set of $G$ , and the connected domination number $γ_{c} (G)$ of $G$ is the smallest number of vertices in a connected dominating set of $G$ . We find the connected domination numbers for all triangulations of up to thirteen vertices. For $n \geq 15$ , $n \equiv 0 (\mod 3)$ , we find graphs of order $n$ and $γ_{c} = n ∕ 3$ . We also show that the difference $γ_{c} (G) - γ (G)$ can be arbitrarily large.

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Keywords

domination, connected domination, triangulation

Mathematical Subject Classification

Primary: 05C10

Milestones

Received: 18 November 2023

Accepted: 18 February 2024

Published: 28 April 2025

Communicated by Joel Foisy

Authors

Felicity Bryant
Department of Mechanical, Aerospace, and Biomedical Engineering University of South Alabama Mobile, AL United States

Elena Pavelescu
Mathematics and Statistics Department University of South Alabama Mobile, AL United States

Vol. 18, No. 3, 2025