Excluding an apex-forest or a fan as quickly as possible

2026-02-03

AI Summary

This study addresses the problem of bounding structural parameters for graphs excluding certain minors, specifically apex-forests or fans. Leveraging graph minor theory together with tools such as layered pathwidth and layered treedepth, we improve upon existing bounds: for graphs excluding an apex-forest $H$, we reduce the upper bound on layered pathwidth to $|V(H)| - 2$; for graphs excluding an apex-linear forest (which includes fans), we lower the upper bound on layered treedepth from quadratic to linear, also achieving $|V(H)| - 2$. Both bounds are tight and optimal, representing a significant improvement over previous results.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

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
• 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 show that every graph $G$ excluding an apex-forest $H$ as a minor has layered pathwidth at most $|V(H)|-2$, and that every graph $G$ excluding an apex-linear forest (such as a fan) $H$ as a minor has layered treedepth at most $|V(H)|-2$. We further show that both bounds are optimal. These results improve on recent results of Hodor, La, Micek, and Rambaud (2025): The first result improves the previous best-known bound by a multiplicative factor of $2$, while the second strengthens a previous quadratic bound. In addition, we reduce from quadratic to linear the bound on the $S$-focused treedepth $\mathrm{td}(G,S)$ for graphs $G$ with a prescribed set of vertices $S$ excluding models of paths in which every branch set intersects~$S$.

Problem

• apex-forest
• fan
• graph minor
• layered pathwidth
• layered treedepth

Innovation

• layered pathwidth
• layered treedepth
• apex-forest
• graph minor
• S-focused treedepth