The Weisfeiler-Leman Algorithm and Recognition of Graph Properties

TL;DR

This paper investigates the capabilities of the Weisfeiler-Leman algorithm in recognizing graph properties like vertex- and arc-transitivity, revealing its limitations depending on graph size and the number of WL dimensions used.

Contribution

The paper demonstrates how the Weisfeiler-Leman algorithm can identify certain graph symmetries in some cases but fails in others, especially for larger graphs with specific divisibility properties.

Findings

2-WL detects vertex-transitivity for prime-sized graphs.

k-WL cannot distinguish transitivity in graphs divisible by 16 with k=o(√n).

Similar limitations apply to arc-transitivity recognition.

Abstract

The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of $k$-WL to recognition of graph properties. Let $G$ be an input graph with $n$ vertices. We show that, if $n$ is prime, then vertex-transitivity of $G$ can be seen in a straightforward way from the output of 2-WL on $G$ and on the vertex-individualized copies of $G$. However, if $n$ is divisible by 16, then $k$-WL is unable to distinguish between vertex-transitive and non-vertex-transitive graphs with $n$ vertices as long as $k=o(\sqrt n)$. Similar results are obtained for recognition of arc-transitivity.