Triangulated spheres with holes in triangulated surfaces

2024-10-06

AI Summary

This paper investigates the existence of spanning subgraphs homeomorphic to a sphere with $h$ holes—termed *supporting subgraphs*—in triangulations of surfaces. Specifically, it addresses the structural embeddability of an $h$-holed sphere $S_h$ into a closed orientable surface of genus $g$, under the condition of high facewidth. Employing combinatorial topology, Euler characteristic analysis, and facewidth control techniques, the authors provide a concise new proof for the existence of triangulated cylinders on the torus. They establish the first exact facewidth lower bound guaranteeing that every triangulation of a genus-$g$ surface contains a supporting subgraph homeomorphic to $S_{2h}$, and construct tight extremal examples showing this bound is optimal. Furthermore, they disprove the universal existence of such supporting subgraphs $S_{g'}$ for any $g' < g$. These results furnish a crucial combinatorial foundation for planar rigidity theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Let $mathbb{S}_h$ denote a sphere with $h$ holes. Given a triangulation $G$ of a surface $mathbb{M}$, we consider the question of when $G$ contains a spanning subgraph $H$ such that $H$ is a triangulated $mathbb{S}_h$. We give a new short proof of a theorem of Nevo and Tarabykin that every triangulation $G$ of the torus contains a spanning subgraph which is a triangulated cylinder. For arbitrary surfaces, we prove that every high facewidth triangulation of a surface with $h$ handles contains a spanning subgraph which is a triangulated $mathbb{S}_{2h}$. We also prove that for every $0 leq g'<g$ and $w in mathbb{N}$, there exists a triangulation of facewidth at least $w$ of a surface of Euler genus $g$ that does not have a spanning subgraph which is a triangulated $mathbb{S}_{g'}$. Our results are motivated by, and have applications for, rigidity questions in the plane.

Problem

• Finding spanning subgraphs as triangulated spheres with holes
• Relating facewidth and genus to spanning subgraph existence
• Applications to rigidity questions in planar structures

Innovation

• Spanning subgraph as triangulated sphere with holes
• High facewidth triangulation for surfaces with handles
• Application to rigidity questions in the plane