Connected Dominating Sets in Triangulations

TL;DR

This paper proves that every n-vertex triangulation has a small connected dominating set of size at most 10n/21, improving previous bounds and enabling applications in graph drawing and surface triangulations.

Contribution

The paper establishes a tighter bound on connected dominating sets in triangulations, extending to surfaces of higher genus, and applies this to graph drawing and geometric representations.

Findings

Connected dominating set size at most 10n/21 in triangulations.

Improved bounds for the SEFENOMAP graph drawing problem.

Extension of results to genus-g surface triangulations.

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 $10n/21+O(\sqrt{gn})$.