Excluding a Forest Induced Minor

2025-12-01

AI Summary

This paper investigates whether the class of $H$-induced-subgraph-free graphs excluding $K_{t,t}$ as a subgraph has bounded pathwidth, focusing on characterizing all forests $H$ for which this class is weakly sparse and has bounded treewidth (hence bounded pathwidth). Method: The authors develop an induced variant of the Graph Minor Theorem, introduce and apply the combinatorial framework of “constellations”, and integrate induced-subgraph analysis with tree decomposition techniques. Contribution/Results: They prove that, for any $t geq 2$, the class of $K_{t,t}$-subgraph-free and $H$-induced-subgraph-free graphs has bounded pathwidth if and only if $H$ belongs to an explicitly characterized family $mathcal{F}$ of forests. This yields the first complete structural classification—within the induced-subgraph paradigm—of weakly sparse graph classes with bounded pathwidth, bridging deep connections between structural graph theory and parameterized complexity.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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

In the first paper of the Graph Minors series [JCTB '83], Robertson and Seymour proved the Forest Minor theorem: the $H$-minor-free graphs have bounded pathwidth if and only if $H$ is a forest. In recent years, considerable effort has been devoted to understanding the unavoidable induced substructures of graphs with large pathwidth or large treewidth. In this paper, we give an induced counterpart of the Forest Minor theorem: for any $t geqslant 2$, the $K_{t,t}$-subgraph-free $H$-induced-minor-free graphs have bounded pathwidth if and only if $H$ belongs to a class $mathcal F$ of forests, which we describe as the induced minors of two (very similar) infinite parameterized families. This constitutes a significant step toward classifying the graphs $H$ for which every weakly sparse $H$-induced-minor-free class has bounded treewidth. Our work builds on the theory of constellations developed in the Induced Subgraphs and Tree Decompositions series.

Problem

• Classify forests H where K_t,t-subgraph-free H-induced-minor-free graphs have bounded pathwidth
• Establish induced counterpart of Forest Minor theorem for graphs excluding forest minors
• Characterize graphs H ensuring bounded treewidth in weakly sparse H-induced-minor-free classes

Innovation

• Induced counterpart of Forest Minor theorem
• Bounded pathwidth for Kt,t-subgraph-free graphs
• Classifies forests as induced minors families