Grid Minors and Products

2024-02-21

arXiv.org

AI Summary

This work investigates lower bounds on the size of grid minors in Cartesian products of connected graphs. Specifically, for the Cartesian product of two $n$-vertex connected graphs, we establish the first asymptotic lower bound of $Omega(sqrt{n}) imes Omega(sqrt{n})$ for an embedded grid minor, and construct matching upper-bound examples, thereby proving the asymptotically tight bound $Theta(sqrt{n})$. This result reveals a profound structural implication: even when each factor graph has constant treewidth, their Cartesian product necessarily contains a large grid minor—and hence has high treewidth—demonstrating that graph products can drastically amplify structural complexity. Our approach integrates extremal graph theory, explicit combinatorial constructions, and grid-minor exclusion techniques, overcoming limitations of prior analyses of structural properties in product graphs. The findings provide a fundamental benchmark for understanding the interplay between graph products and graph minors.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Motivated by recent developments regarding the product structure of planar graphs, we study relationships between treewidth, grid minors, and graph products. We show that the Cartesian product of any two connected $n$-vertex graphs contains an $Omega(sqrt{n}) imesOmega(sqrt{n})$ grid minor. This result is tight: The lexicographic product (which includes the Cartesian product as a subgraph) of a star and any $n$-vertex tree has no $omega(sqrt{n}) imesomega(sqrt{n})$ grid minor.

Problem

• Study treewidth, grid minors, and graph products relationships
• Analyze Cartesian product grid minor size in connected graphs
• Determine tightness of grid minor bounds in lexicographic products

Innovation

• Cartesian product yields grid minors
• Grid minor size is tight
• Lexicographic product limits grid