The Weisfeiler-Leman dimension of distance-hereditary graphs

TL;DR

This paper proves that the Weisfeiler-Leman algorithm can correctly test isomorphism for distance-hereditary graphs using only 2-dimensional refinement, significantly improving the previous upper bound of 7.

Contribution

It establishes that the Weisfeiler-Leman dimension for distance-hereditary graphs is exactly 2, advancing understanding of graph isomorphism testing.

Findings

Weisfeiler-Leman algorithm correctly tests isomorphism for distance-hereditary graphs.

The Weisfeiler-Leman dimension for this class is exactly 2.

Previous upper bound was 7, now improved to 2.

Abstract

A graph is said to be distance-hereditary if the distance function in every connected induced subgraph is the same as in the graph itself. We prove that the ordinary Weisfeiler-Leman algorithm correctly tests the isomorphism of any two graphs if one of them is distance-hereditary; more precisely, the Weisfeiler-Leman dimension of the class of finite distance-hereditary graphs is equal to $2$. The previously best known upper bound for the dimension was $7$.