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arXiv · 2312.03399

Connected Dominating Sets in Triangulations

Prosenjit Bose·Vida Dujmović·Hussein Houdrouge·Pat Morin·Saeed Odak
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Abstract

We show that every $n$-vertex triangulation has a connected dominating set of size at most $10n/21$. Equivalently, every $n$ vertex triangulation has a spanning tree with at least $11n/21$ leaves. Prior to the current work, the best known bounds were $n/2$, which follows from work of Albertson, Berman, Hutchinson, and Thomassen (J. Graph Theory \textbf{14}(2):247--258). One immediate consequence of this result is an improved bound for the SEFENOMAP graph drawing problem of Angelini, Evans, Frati, and Gudmundsson (J. Graph Theory \textbf{82}(1):45--64). As a second application, we show that for every set $P$ of $\lceil 11n/21\rceil$ points in $\R^2$ every $n$-vertex planar graph has a one-bend non-crossing drawing in which some set of $11n/21$ vertices is drawn on the points of $P$. The main result extends to $n$-vertex triangulations of genus-$g$ surfaces, and implies that these have connected dominating sets of size at most $10n/21+O(\sqrt{gn})$.

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Prosenjit Bose, Vida Dujmović, Hussein Houdrouge, Pat Morin, Saeed Odak. 2023-12-06. Connected Dominating Sets in Triangulations. https://arxiv.org/abs/2312.03399

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