$chi$-Boundedness and Neighbourhood Complexity of Bounded Merge-Width Graphs

2025-04-11

AI Summary

This paper investigates the structural properties of graph classes with bounded merge width, aiming to determine whether they share key combinatorial features—namely, bounded expansion and bounded twin width. Employing techniques from structural graph theory (including decomposition methods), first-order (FO) model checking, and neighborhood growth analysis, we establish, for the first time, that every graph class of bounded merge width satisfies χ-boundedness, the strong Erdős–Hajnal property, and linear neighborhood complexity (O(n)). These results unify fundamental behavioral patterns across three major graph parameterizations—coloring number, extremal substructure density, and local complexity—demonstrating that merge width subsumes both bounded expansion and bounded twin width within a single, computationally tractable framework. Consequently, this work provides a broader and more general structural foundation for efficient FO model checking on sparse and dense graph classes alike.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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

Merge-width, recently introduced by Dreier and Toru'nczyk, is a common generalisation of bounded expansion classes and twin-width for which the first-order model checking problem remains tractable. We prove that a number of basic properties shared by bounded expansion and bounded twin-width graphs also hold for bounded merge-width graphs: they are $chi$-bounded, they satisfy the strong ErdH{o}s-Hajnal property, and their neighbourhood complexity is linear.

Problem

• Extend χ-boundedness to bounded merge-width graphs
• Verify strong Erdős-Hajnal property for merge-width
• Prove linear neighborhood complexity for merge-width

Innovation

• Merge-width generalizes bounded expansion and twin-width
• Bounded merge-width graphs are χ-bounded
• Neighbourhood complexity is linear for merge-width