Core Concepts

Every n-vertex triangulation has a connected dominating set of size at most 10n/21.

Abstract

The paper presents algorithms for efficiently constructing small connected dominating sets in triangulations. The key results are:
Every n-vertex triangulation has a connected dominating set of size at most 10n/21. This improves upon the previous best bound of n/2.
Every n-vertex triangulation has a spanning tree with at least 11n/21 leaves. This provides progress on the maxleaf spanning-tree problem for triangulations.
The results extend to n-vertex triangulations of genus-g surfaces, where the connected dominating set size is at most 10n/21 + O(√gn).
The algorithms work by incrementally growing a connected dominating set in small batches. The main technical challenges are:
Identifying and handling "critical" subgraphs where vertices have inner-degree at most 2.
Efficiently dominating degree-1 and degree-0 vertices in the critical subgraphs.
Carefully constructing a sequence of sets that satisfy the properties required for the final connected dominating set.
The paper also discusses connections to the problem of finding one-bend free sets in planar graphs, showing that the leaves of the spanning tree provide a one-bend free set of size at least 11n/21.

Stats

Every n-vertex triangulation has a dominating set of size at most n/3.
Every n-vertex triangulation has a connected dominating set of size at most 2n/3.
Every n-vertex triangulation has a spanning tree with at least (n+2)/2 leaves.

Quotes

"For every n ≥3, every n-vertex triangulation G has a connected dominating set X of size at most 10n/21 = 0.476190n."
"For every n ≥3, every n-vertex triangulation G has a spanning tree T with at least 11n/21 = 0.523809n leaves."

Deeper Inquiries

The techniques developed in this paper can be extended to other families of graphs beyond triangulations by adapting the algorithmic approach to suit the specific characteristics of those graphs. For instance, for graphs with higher minimum degree requirements or different structural constraints, the algorithm can be modified to ensure the construction of connected dominating sets within those constraints. Additionally, the concept of dom-minimal graphs and dom-respecting subgraphs can be applied to various graph families to optimize the process of finding connected dominating sets.

The results presented in this paper have significant implications for practical applications in network design and graph visualization. In network design, the ability to efficiently find connected dominating sets in triangulations can lead to more robust and efficient network structures. These sets can be utilized for tasks such as routing, broadcasting, and fault tolerance in communication networks. In graph visualization, the concept of one-bend free sets and non-crossing drawings can enhance the clarity and aesthetics of graph representations, making them more understandable for users.

There are several connections between the problems studied in this paper and other fundamental problems in graph theory and combinatorial optimization. The investigation of connected dominating sets in triangulations is closely related to the broader study of dominating sets in graphs, which is a fundamental topic in graph theory. The development of efficient algorithms for finding connected dominating sets can contribute to the advancement of algorithmic graph theory and optimization. Additionally, the exploration of one-bend free sets and non-crossing drawings has implications for graph drawing algorithms and geometric graph theory, connecting the research to the field of computational geometry.

0