The Weisfeiler–Leman Dimension of Distance-Hereditary Graphs

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 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.