Core Concepts

The authors demonstrate the existence of dominating and face-hitting sets in plane graphs, highlighting the necessity of specific conditions for their conclusions.

Abstract

The content explores the concept of dominating sets and face-hitting sets in planar graphs. It discusses the partitioning of vertex sets into two disjoint subsets that are both dominating and face-hitting. The analysis includes corollaries related to simple plane triangulations and conjectures by Matheson and Tarjan. The paper also delves into polychromatic colorings, proving theorems, lemmas, and observations to support the main arguments. Furthermore, it provides applications to Matheson-Tarjan Conjecture and references previous works on domination numbers in planar graphs.

Stats

Every n-vertex simple plane triangulation has a dominating set of size at most (1 − α)n/2.
Currently, every plane triangulation on n > 10 vertices has a dominating set of size at most 2n/7.
For triangulated discs, the upper bound of n/3 by Matheson and Tarjan is tighter than general bounds.
Fomin and Thilikos showed that the k-dominating set problem on planar graphs can be solved in time O(215.13√k +n3).
Botler et al. proved that every planar triangulation G on n vertices has an independent dominating set of size less than 3n/8.

Quotes

"Every plane graph without isolated vertices, self-loops or 2-faces has a 2-coloring which is simultaneously domatic and polychromatic." - Theorem 1
"Our corollary improves their bound for n-vertex plane triangulations which contain a maximal independent set of size either less than 2n/7 or more than 3n/7." - Content Summary

Key Insights Distilled From

by P. Francis,A... at **arxiv.org** 03-06-2024

Deeper Inquiries

Face-hitting sets have a direct connection to terrain guarding problems. In the context of graph theory, face-hitting sets in plane graphs can be linked to guarding polyhedral terrains. The idea is that by identifying specific vertices within a plane graph that form a face-hitting set, one can draw parallels to strategically placing guards or sensors on a polyhedral terrain to ensure comprehensive coverage and surveillance. This concept allows for the optimization of resources and efficient monitoring strategies in real-world scenarios.

The research on face-hitting dominating sets in planar graphs extends beyond theoretical mathematics and holds practical implications in various fields. One significant application lies in network security, where these findings can be utilized to enhance intrusion detection systems. By identifying optimal face-hitting dominating sets within network topologies, vulnerabilities can be effectively monitored and mitigated, leading to improved cybersecurity measures.
Furthermore, these results have implications in urban planning and facility management. By applying the principles of dominating sets derived from this research, city planners can optimize resource allocation for services such as waste collection or public transportation routes. Similarly, facility managers can use these concepts to streamline maintenance schedules based on critical areas identified through dominating sets analysis.

The insights gained from studying face-hitting dominating sets in planar graphs are not limited solely to this specific type of graph but can also be applied more broadly across different graph structures. For instance:
General Graphs: The concept of dominating sets transcends planar graphs and can be extended to general graphs with varying degrees of complexity.
Social Networks: Analyzing social networks using dominating set principles could help identify influential individuals or groups crucial for information dissemination or opinion formation.
Biological Networks: In biological networks like protein-protein interaction networks, understanding dominant nodes through similar mechanisms could reveal key proteins essential for cellular functions.
Telecommunication Networks: Applying domination concepts in telecommunication networks may aid in optimizing signal transmission paths or enhancing network reliability by identifying critical nodes for redundancy planning.
By adapting the methodologies developed for planar graphs into these diverse applications, researchers and practitioners stand to benefit from enhanced problem-solving capabilities across multiple domains utilizing graph theory principles related to dominance structures.

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