# A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus

## TL;DR

This paper establishes a linear upper bound on the Weisfeiler-Leman dimension for graphs embeddable on surfaces of bounded genus, improving understanding of their structural complexity.

## Contribution

It provides the first linear upper bounds on WL dimension for graphs of bounded genus, extending previous results known for planar graphs.

## Key findings

- WL dimension of graphs on surfaces of genus g is at most 4g+3
- WL dimension of graphs on orientable surfaces of genus g is at most 2g+3
- Boundaries are linear in the genus g

## Abstract

The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be characterised in terms of the number of variables that is required to describe the graph up to isomorphism in first-order logic with counting quantifiers. It is known that the WL dimension is upper-bounded for all graphs that exclude some fixed graph as a minor (Grohe, JACM 2012). However, the bounds that can be derived from this general result are astronomic. Only recently, it was proved that the WL dimension of planar graphs is at most 3 (Kiefer, Ponomarenko, and Schweitzer, LICS 2017). In this paper, we prove that the WL dimension of graphs embeddable in a surface of Euler genus $g$ is at most $4g+3$. For the WL dimension of graphs embeddable in an orientable surface of Euler genus $g$, our approach yields an upper bound of $2g+3$.