Connected Dominating Sets in Triangulations

TL;DR

This work improves the upper bound on the size of connected dominating sets in $n$-vertex triangulations to $\frac{10n}{21}$, equivalently yielding a spanning tree with at least $\frac{11n}{21}$ leaves. The authors develop a refined, linear-time algorithm (BetterGreedy) built on an outer-domatic framework and a detailed analysis of critical graphs, dom-minimal substitutions, and 2-critical cases, advancing from the prior $\frac{n}{2}$ bound. They extend the result to Euler genus $g$ surface triangulations with an additive $O(\sqrt{gn})$ term and apply the bound to graph drawing, showing that a spanning tree with many leaves induces a large one-bend free/collinear set, with consequences for SEFE problems and planar drawings. The paper also develops a genus-aware slicing approach and demonstrates how the leaves of a spanning tree yield one-bend collinear sets, connecting domination theory to geometric representations. Overall, the approach provides both a concrete combinatorial improvement and practical algorithms with broad implications in graph drawing and planar/topological graph theory.

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

Figures (19)

• Figure 1: Some critical graphs.
• Figure 2: Three cases on the way to making $H$ dom-minimal.
• Figure 3: The proof of \ref{['chord_incident']}
• Figure 4: The proof of \ref{['degree_2_outer_neighbour']}.
• Figure 5: Removing an inner-degree $2$ vertex $v$ is immediately followed by removing an inner-degree $3$ vertex $w$.

Theorems & Definitions (59)

• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3