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The Weisfeiler–Leman Dimension of Distance-Hereditary Graphs

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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 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.

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Notes

  1. A matching graph in the terminology of [1].

  2. In fact, for the “if” part, Condition (1) already suffices.

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Acknowledgements

The research of Alexander Gavrilyuk is supported by JSPS KAKENHI Grant Number 22K03403. Roman Nedela is supported by the Czech Science Foundation, grant GACR 20-15576S and by the Slovak Research and Development Agency, Grant No. APVV-15-0220.

Funding

Japan Society for the Promotion of Science London, 22K03403, Alexander Gavrilyuk. Czech Science Foundation, GACR 20-15576S, Roman Nedela. Slovak Research and Development Agency, APVV-15-0220, Roman Nedela.

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Authors and Affiliations

  1. Shimane University, Matsue, Japan

    Alexander L. Gavrilyuk

  2. Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic

    Roman Nedela

  3. School of Mathematics and Statistics of Central China Normal University, Wuhan, China

    Ilia Ponomarenko

  4. St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia

    Ilia Ponomarenko

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  1. Alexander L. Gavrilyuk
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  2. Roman Nedela
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Correspondence to Alexander L. Gavrilyuk.

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Gavrilyuk, A.L., Nedela, R. & Ponomarenko, I. The Weisfeiler–Leman Dimension of Distance-Hereditary Graphs. Graphs and Combinatorics 39, 84 (2023). https://doi.org/10.1007/s00373-023-02683-3

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