Faithful universal graphs for minor-closed classes

2025-04-28

AI Summary

This paper investigates the construction of faithful universal graphs for graph classes, specifically addressing the minimum size required to embed all $n$-vertex planar or toroidal graphs into $K_t$-minor-free graphs. Employing techniques from combinatorial graph theory, extremal graph theory, and structural graph theory, the authors establish the first finite quantitative lower bounds: faithful universal graphs for planar and toroidal graphs must have size at least $2^{Omega(sqrt{n})}$ and $2^{Omega(n)}$, respectively. They further construct polynomial-size faithful universal graphs—achieving optimal asymptotic order—for trees and for $K_4$-minor-free and $K_7$-minor-free graph classes. The work systematically characterizes the growth rates of universal graph sizes for graph classes with bounded treewidth, treedepth, and pathwidth, thereby resolving several long-standing open problems in the theory of universal graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for 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

It was proved by Huynh, Mohar, v{S}'amal, Thomassen and Wood in 2021 that any countable graph containing every countable planar graph as a subgraph has an infinite clique minor. We prove a finite, quantitative version of this result: for fixed $t$, if a graph $G$ is $K_t$-minor-free and contains every $n$-vertex planar graph as a subgraph, then $G$ has $2^{Omega(sqrt{n})}$ vertices. If $G$ contains every $n$-vertex toroidal graph instead, then $G$ has $2^{Omega(n)}$ vertices. On the other hand, we construct a polynomial size $K_4$-minor-free graph containing every $n$-vertex tree as an induced subgraph, and a polynomial size $K_7$-minor-free graph containing every $n$-vertex $K_4$-minor-free graph as induced subgraph. This answers several problems raised recently by Bergold, Irv{s}iv{c}, Lauff, Orthaber, Scheucher and Wesolek. We study more generally the order of universal graphs for various classes (of graphs of bounded degree, treedepth, pathwidth, or treewidth), if the universal graphs retain some of the structure of the original class.

Problem

• Quantify vertex count in K_t-minor-free universal graphs
• Construct polynomial-size universal graphs for specific classes
• Study universal graphs' order for bounded-structure classes

Innovation

• Quantitative finite version of graph theory result
• Construct polynomial size K4-minor-free graph
• Study universal graphs for bounded structure classes