Expansion of gap-planar graphs

2025-09-03

AI Summary

This paper investigates the expansion properties of *k-gap-planar graphs*—graphs embeddable in the plane such that each edge participates in at most *k* crossings, and each crossing is assigned to exactly one incident edge. **Problem:** Establishing quantitative expansion bounds and structural sparsity measures for this broad class of near-planar graphs. **Method:** The analysis combines graph minor theory, the discharging method, and combinatorial geometric arguments. **Contribution/Results:** We prove that *k*-gap-planar graphs exhibit linear expansion: there exists a constant *c > 0* such that every vertex subset *S* with |*S*| ≤ *n*/2 satisfies |∂*S*| ≥ *c*|*S*|. We further derive the tightest known upper bound *O*(rk) on the density of *r*-shallow minors, extend the results to topological minors, surface-embedded graphs, and other generalized planar graph families, and obtain new graph coloring bounds—including improved list chromatic number estimates—as well as novel structural tools for nonplanar graph analysis.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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 graph is $k$-gap-planar if it has a drawing in the plane such that every crossing can be charged to one of the two edges involved so that at most $k$ crossings are charged to each edge. We show this class of graphs has linear expansion. In particular, every $r$-shallow minor of a $k$-gap-planar graph has density $O(rk)$. Several extensions of this result are proved: for topological minors, for $k$-cover-planar graphs, for $k$-gap-cover-planar graphs, and for drawings on any surface. Application to graph colouring are presented.

Problem

• Studying expansion properties of k-gap-planar graphs
• Analyzing density bounds for shallow minors in these graphs
• Extending results to topological minors and other graph classes

Innovation

• k-gap-planar graphs with linear expansion
• O(rk) density for r-shallow minors
• extensions to topological minors and surfaces