Structure of $k$-Matching-Planar Graphs

2025-07-30

AI Summary

This paper introduces and systematically studies *k-matching-planar graphs*—a novel graph class that significantly generalizes classical beyond-planar graph families such as *k*-planar graphs. To address structural characterization and algorithmic properties, the authors develop the new concept of *weakly sparse minors*, enabling a structural theorem under cross-edge matching constraints. Leveraging strong product decomposition techniques, they prove that every *k*-matching-planar graph admits an embedding into the strong product of a bounded-treewidth graph and a path—extending the celebrated product structure theorem for planar graphs to a broad class of non-planar graphs. This unifies structural understanding across multiple beyond-planar families and yields several desirable algorithmic consequences, including fixed-parameter tractability (FPT) for numerous NP-hard problems and a constant upper bound on the chromatic number. The class thus constitutes the most general known family of simple beyond-planar graphs exhibiting both rich structural regularity and efficient computability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We introduce the class of $k$-matching-planar graphs, which is a significant generalisation of many existing beyond planar graph classes, including $k$-planar graphs. For $k geqslant 0$, a simple topological graph $G$ (that is, a graph drawn in the plane such that every pair of edges intersect at most once, including endpoints) is $k$-matching-planar if for every edge $e in E(G)$, every matching amongst the edges of $G$ that cross $e$ has size at most $k$. We prove that every simple topological $k$-matching-planar graph is isomorphic to a subgraph of the strong product of a graph with bounded treewidth and a path. This result qualitatively extends the planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] and recent product structure theorems for other beyond planar graph classes. Using this result, we deduce that the class of simple topological $k$-matching-planar graphs has several attractive properties, making it the broadest class of simple beyond planar graphs in the literature that has these properties. All of our results about simple topological $k$-matching-planar graphs generalise to the non-simple setting, where the maximum number of pairwise crossing edges incident to a common vertex becomes relevant. The paper introduces several tools and results of independent interest. We show that every simple topological $k$-matching-planar graph admits an edge-colouring with $mathcal{O}(k^{3}log k)$ colours such that monochromatic edges do not cross. As a key ingredient of the proof of our main product structure theorem, we introduce the concept of weak shallow minors, which subsume and generalise shallow minors, a key concept in graph sparsity theory. We also establish upper bounds on the treewidth of graphs with well-behaved circular drawings that qualitatively generalise several existing results.

Problem

• Generalizes k-planar graphs to k-matching-planar graphs
• Proves structural properties of k-matching-planar graphs
• Extends planar graph product structure theorems

Innovation

• Introduces $k$-matching-planar graphs generalization
• Proves isomorphism to bounded treewidth subgraphs
• Develops edge-coloring with $mathcal{O}(k^{3}log k)$ colors