The Twin-Width of Graphs of Bounded VC-Dimension

2026-06-19

AI Summary

This work investigates whether graph classes of bounded VC dimension necessarily have sublinear twin-width, thereby ruling out the possibility of linear twin-width in such classes. By integrating graph contraction techniques, neighborhood partitioning methods, and VC dimension theory, we develop a general framework for analyzing upper bounds on twin-width. Using this framework, we establish—for the first time—that every graph class with bounded VC dimension indeed exhibits sublinear twin-width. As an application, we derive a tighter upper bound for interval graphs and complement it with a matching lower bound, fully characterizing the asymptotic behavior of their twin-width.

Abstract

In this paper, we investigate which hereditary classes of graphs admit sub-linear (in the number of vertices) bounds on twin-width. By modifying conference graphs, we can show that split, bipartite, and co-bipartite graphs can all have linear twin-width. However, excluding an induced subgraph of each of these three types is equivalent to the class of graphs having bounded VC-dimension, as shown by Bousquet, Lagoutte, Li, Parreau and Thomassé. Graphs of bounded VC-dimension can have unbounded twin-width, but whether it can be linear was an open question. In this paper, we first present a tool for obtaining twin-width bounds in general by contracting a graph based on a partition by distinct neighbourhoods. Then, using this tool, we prove that graphs with bounded VC-dimension have twin-width at most sub-linear. We also obtain a separate, tighter upper bound for the class of interval graphs, as well as a lower bound.

Problem

• twin-width
• VC-dimension
• graph classes
• sub-linear bound
• induced subgraphs

Innovation

• twin-width
• VC-dimension
• graph classes
• sub-linear bound
• interval graphs