On the size of k-irreducible triangulations

2026-03-20

AI Summary

This study addresses the problem of determining a tight upper bound on the number of triangles in a $k$-irreducible triangulation of a surface of genus $g$. By integrating techniques from combinatorial topology, graph embeddings on surfaces, edge-contraction analysis, and estimates on the lengths of non-contractible curves, the authors establish a precise relationship between the number of triangles and the parameters $k$ and $g$. The main contribution is a significant improvement over the previously known bound of $k^{O(k)} g^2$, yielding a tight and asymptotically optimal bound of $O(k^2 g)$. This result elucidates the fundamental dependence of the size of such triangulations on both the irreducibility parameter $k$ and the topological complexity of the underlying surface as measured by its genus.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of a surface of genus g has $O(k^2g)$ triangles, which is optimal. This is an improvement over the previous best bound $k^{O(k)} g^2$ of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].

Problem

• k-irreducible triangulations
• non-contractible curves
• surface genus
• triangle complexity

Innovation

• k-irreducible triangulation
• non-contractible curve
• surface genus
• edge contraction
• combinatorial upper bound