Variants of Merge-Width and Applications

TL;DR

This paper explores various definitions and properties of merge-width, a graph parameter that unifies several width measures, and introduces new characterizations, algorithms, and connections to other graph concepts.

Contribution

It proves the equivalence of multiple definitions of merge-width, introduces a non-trivial approximation algorithm, and establishes new characterizations and properties of graphs with bounded merge-width.

Findings

Proved equivalence of alternative merge-width definitions

Developed the first non-trivial approximation algorithm for merge-width

Established connections between merge-width and graph sparsity, quasi-isometry, and neighbourhood covers

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.