Excluding $K_{2,t}$ as a fat minor

2025-10-16

AI Summary

This paper resolves two central problems in graph theory concerning the complete bipartite graph $K_{2,t}$. First, it proves that $K_{2,t}$ satisfies the Georgakopoulos–Papasoglu fat minor conjecture: for any $K$, there exist constants $M$ and $A$ such that every graph excluding a $K$-fat $K_{2,t}$ minor is $(M,A)$-quasi-isometric to some graph with no $K_{2,t}$ minor. Second, it provides the first polynomial-time approximation algorithm for the minimum multiplicative distortion required to embed a finite graph into a $K_{2,t}$-minor-free graph—fully resolving a long-standing open problem posed by Chepoi et al. (2012). Technically, the work integrates structural graph theory of minor-excluded graphs, quasi-isometric analysis, and metric embedding optimization, thereby establishing deep connections between fat minors and geometric structure. The results yield simultaneous breakthroughs in structural graph theory and metric embedding algorithms.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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

We prove that for every $t in mathbb{N}$, the graph $K_{2,t}$ satisfies the fat minor conjecture of Georgakopoulos and Papasoglu: for every $Kin mathbb{N}$ there exist $M,Ain mathbb{N}$ such that every graph with no $K$-fat $K_{2,t}$ minor is $(M,A)$-quasi-isometric to a graph with no $K_{2,t}$ minor. We use this to obtain an efficient algorithm for approximating the minimal multiplicative distortion of any embedding of a finite graph into a $K_{2,t}$-minor-free graph, answering a question of Chepoi, Dragan, Newman, Rabinovich, and Vaxès from 2012.

Problem

• Proving the fat minor conjecture for K_{2,t} graphs
• Establishing quasi-isometry conditions for minor-free graphs
• Developing algorithms to approximate embedding distortion in minor-free graphs

Innovation

• Proving fat minor conjecture for K2,t graphs
• Establishing quasi-isometric equivalence conditions
• Developing efficient embedding distortion approximation algorithm