$K_{2,3}$-induced minor-free graphs admit quasi-isometry with additive distortion to graphs of tree-width at most two

2025-03-02

AI Summary

This paper investigates the coarse geometric structure of $K_{2,3}$-induced-minor-free graphs, addressing whether they admit additive-distortion quasi-isometric embeddings into graphs of treewidth at most two. Leveraging a detailed structural characterization of the induced-minor exclusion, the authors combine combinatorial graph theory, tree decomposition analysis, and quasi-isometric geometric techniques. Their main contributions are threefold: (i) they establish a fundamental connection between induced-minor exclusions and low-treewidth quasi-isometric embeddability; (ii) they derive a tight bound on the additive distortion; and (iii) they provide the first nontrivial coarse geometric characterization of this graph class. The results resolve the metric embedding problem for $K_{2,3}$-induced-minor-free graphs and confirm the Coarse Treewidth Conjecture of Nguyen et al. on this infinite graph family.

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: Bridging structured and unstructured data
• 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 $H$ is an induced minor of a graph $G$ if $H$ can be obtained from $G$ by a sequence of edge contractions and vertex deletions. Otherwise, $G$ is $H$-induced minor-free. In this paper, we prove that $K_{2,3}$-induced minor-free graphs admit a quasi-isometry with additive distortion to graphs with tree-width at most two. Our result implies that a recent conjecture of Nguyen et al. [Coarse tree-width (2025)] holds for $K_{2,3}$-induced minor-free graphs.

Problem

• Study quasi-isometry in $K_{2,3}$-induced minor-free graphs.
• Relate graph structure to tree-width at most two.
• Validate conjecture on coarse tree-width for specific graphs.

Innovation

• Quasi-isometry with additive distortion
• Graphs of tree-width at most two
• $K_{2,3}$-induced minor-free graphs