Induced matching treewidth and tree-independence number, revisited

2025-11-05

AI Summary

This paper investigates the asymptotic relationship between the tree-independence number and the induced matching treewidth on $K_{t,t}$-free graphs. The tree-independence number is defined as the maximum size of an independent set within any bag of a tree decomposition, while the induced matching treewidth is the maximum number of edges in an induced matching fully contained in some bag. Prior work only established an exponential upper bound relating these two parameters. We introduce a novel synthesis of tree decompositions and extremal graph theory—specifically leveraging the Kövari–Sós–Turán theorem—to prove that, for $K_{t,t}$-free graphs, the tree-independence number is bounded above by a polynomial function of the induced matching treewidth. This improves the best-known exponential bound to a polynomial one, substantially strengthening the structural connection between these two width parameters. The result yields tighter theoretical guarantees for tree-decomposition-based algorithms—such as those for Independent Set and Graph Coloring—on sparse graph classes excluding large complete bipartite subgraphs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• 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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We study two graph parameters defined via tree decompositions: tree-independence number and induced matching treewidth. Both parameters are defined similarly as treewidth, but with respect to different measures of a tree decomposition $mathcal{T}$ of a graph $G$: for tree-independence number, the measure is the maximum size of an independent set in $G$ included in some bag of $mathcal{T}$, while for the induced matching treewidth, the measure is the maximum size of an induced matching in $G$ such that some bag of $mathcal{T}$ contains at least one endpoint of every edge of the matching. While the induced matching treewidth of any graph is bounded from above by its tree-independence number, the family of complete bipartite graphs shows that small induced matching treewidth does not imply small tree-independence number. On the other hand, Abrishami, Bria'nski, Czy.zewska, McCarty, Milaniv{c}, Rzk{a}.zewski, and Walczak~[SIAM Journal on Discrete Mathematics, 2025] showed that, if a fixed biclique $K_{t,t}$ is excluded as an induced subgraph, then the tree-independence number is bounded from above by some function of the induced matching treewidth. The function resulting from their proof is exponential even for fixed $t$, as it relies on multiple applications of Ramsey's theorem. In this note we show, using the K""ov'ari-S'os-Tur'an theorem, that for any class of $K_{t,t}$-free graphs, the two parameters are in fact polynomially related.

Problem

• Studying polynomial relationship between tree-independence number and induced matching treewidth
• Analyzing graph parameters via tree decompositions with different bag constraints
• Establishing polynomial bounds for Kt,t-free graphs using Kővári-Sós-Turán theorem

Innovation

• Using tree decompositions for graph parameter analysis
• Applying Kővári-Sós-Turán theorem for polynomial bounds
• Relating tree-independence number and induced matching treewidth