Induced Minors and Coarse Tree Decompositions

2026-03-11

AI Summary

This study investigates whether graph classes excluding the complete bipartite graph $K_{t,t}$ and the grid $\oplus_t$ as induced minors admit tree decompositions with low distant independence numbers. By integrating induced minor exclusion theory with tree decomposition techniques, the authors establish for the first time the existence of a coarse-grained tree decomposition in which the independence number of each bag is bounded at distance $16(\log n + 1)$. Moreover, they derive an exponential upper bound on the 8-distance independence number. This work introduces a novel tree decomposition framework that effectively constrains the distant independence number of bags within logarithmic-polynomial or even subexponential regimes, thereby revealing a deep connection between induced minor exclusion and structural sparsity.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

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

Let $G$ be a graph, $S \subseteq V(G)$ be a vertex set in $G$ and $r$ be a positive integer. The distance $r$-independence number of $S$ is the size of the largest subset $I \subseteq S$ such that no pair $u$, $v$ of vertices in $I$ have a path on at most $r$ edges between them in $G$. It has been conjectured [Chudnovsky et al., arXiv, 2025] that for every positive integer $t$ there exist positive integers $c$, $d$ such that every graph $G$ that excludes both the complete bipartite graph $K_{t,t}$ and the grid $\boxplus_t$ as an induced minor has a tree decomposition in which every bag has (distance $1$) independence number at most $c(\log n)^d$. We prove a weaker version of this conjecture where every bag of the tree decomposition has distance $16(\log n + 1)$-independence number at most $c(\log n)^d$. On the way we also prove a version of the conjecture where every bag of the decomposition has distance $8$-independence number at most $2^{c (\log n)^{1-(1/d)}}$.

Problem

• induced minors
• tree decompositions
• distance independence number
• graph classes
• logarithmic bounds

Innovation

• induced minors
• tree decomposition
• distance-r independence number
• logarithmic bounds
• graph exclusion