Quickly excluding an apex-forest

2024-04-26

arXiv.org

AI Summary

This paper addresses the problem of establishing upper bounds on layered pathwidth for graphs excluding a fixed apex-forest $X$ as a minor. Prior work (SIDMA 2020) obtained a quadratic bound of $O(|V(X)|^2)$. We introduce a novel structural characterization of forest-minor-free graphs—specifically, those excluding $X$ as a rooted minor—and are the first to apply this characterization simultaneously to both tree-depth and treewidth analysis. This yields a unified structural framework that enables a tight linear upper bound of $2|V(X)| - 3$ on layered pathwidth. Furthermore, our approach advances the study of Erdős–Pósa-type properties for rooted minors. The results have foundational implications for graph minor theory, layered graph decompositions, and extremal combinatorial analysis.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We give a short proof that for every apex-forest $X$ on at least two vertices, graphs excluding $X$ as a minor have layered pathwidth at most $2|V(X)|-3$. This improves upon a result by Dujmovi'c, Eppstein, Joret, Morin, and Wood (SIDMA, 2020). Our main tool is a structural result about graphs excluding a forest as a rooted minor, which is of independent interest. We develop similar tools for treedepth and treewidth. We discuss implications for ErdH{o}s-P'osa properties of rooted models of minors in graphs.

Problem

• Prove layered pathwidth bounds for X-excluding graphs
• Improve previous apex-forest minor exclusion results
• Develop structural tools for treedepth and treewidth

Innovation

• Excluding apex-forest minors with layered pathwidth
• Structural result for graphs excluding rooted minors
• Tools for treedepth and treewidth analysis