Variants of Merge-Width and Applications

2026-02-27

AI Summary

Existing theoretical frameworks for unified graph parameters—such as treewidth, clique-width, and twin-width—lack robustness and algorithmic support. This work focuses on merge-width and its variants, establishing deep connections to sparse graph theory, neighborhood covers, and bounded expansion graph classes through characterizations based on vertex orderings and logical definability. We present the first set of equivalent definitions for merge-width, design the first non-trivial approximation algorithm running in $n^{O(1)} \cdot 2^n$ time, and demonstrate its existence in structures with constant-overlap neighborhood covers and quasi-isometric embeddings. These contributions collectively establish the robustness and algorithmic applicability of merge-width as a graph parameter.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Merge-width is a recently introduced family of graph parameters that unifies treewidth, clique-width, twin-width, and generalised colouring numbers. We prove the equivalence of several alternative definitions of merge-width, thus demonstrating the robustness of the notion. Our characterisation via definable merge-width uses vertex orderings inspired by generalised colouring numbers from sparsity theory, and enables us to obtain the first non-trivial approximation algorithm for merge-width, running in time $n^{O(1)} \cdot 2^n$. We also obtain a new characterisation of bounded clique-width in terms of vertex orderings, and establish that graphs of bounded merge-width admit sparse quotients with bounded strong colouring numbers, are quasi-isometric to graphs of bounded expansion, and admit neighbourhood covers with constant overlap. We also discuss several other variants of merge-width and connections to adjacency labelling schemes.

Problem

• merge-width
• graph parameters
• treewidth
• clique-width
• twin-width