Twin-width of graphs on surfaces

TL;DR

This paper establishes tight bounds on the twin-width of graphs embedded on surfaces of genus g, providing both theoretical limits and an efficient algorithm to find contraction sequences, advancing understanding of graph structure.

Contribution

It proves an asymptotically optimal upper bound on twin-width for surface-embeddable graphs and introduces a stronger product structure theorem with algorithmic implications.

Findings

Twin-width of surface-embeddable graphs is bounded by 18√47g + O(1).

Provides a quadratic time algorithm to find contraction sequences.

Strengthens the product structure theorem for graphs on surfaces.

Abstract

Twin-width is a width parameter introduced by Bonnet, Kim, Thomass\'e and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. We prove that the twin-width of every graph embeddable in a surface of Euler genus $g$ is $18\sqrt{47g}+O(1)$, which is asymptotically best possible as it asymptotically differs from the lower bound by a constant multiplicative factor. Our proof also yields a quadratic time algorithm to find a corresponding contraction sequence. To prove the upper bound on twin-width of graphs embeddable in surfaces, we provide a stronger version of the Product Structure Theorem for graphs of Euler genus $g$ that asserts that every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition with all bags of size at most eight with a single exceptional bag of size $\max\{8,32g-27\}$.