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arXiv · 2602.23825

Local Equivalence Classes of Distance-Hereditary Graphs using Split Decompositions

Nicholas Connolly·Shin Nishio·Kae Nemoto
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Abstract

Local complement is a graph operation formalized by Bouchet which replaces the neighborhood of a chosen vertex with its edge-complement. This operation induces an equivalence relation on graphs; determining the size of the resulting equivalence classes is a challenging problem in general. Bouchet obtained formulas only for paths and cycles, and brute-force methods are limited to very small graphs. In this work, we extend these results by deriving explicit formulas for several broad families of distance-hereditary graphs, including complete multipartite graphs, clique-stars, and repeater graphs. Our approach uses a technique known as split decomposition to establish upper bounds on equivalence class sizes, and we prove these bounds are tight through a combinatorial enumeration of the graphs' decomposed structure up to symmetry.

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BibTeXRIS

Nicholas Connolly, Shin Nishio, Kae Nemoto. 2026-02-27. Local Equivalence Classes of Distance-Hereditary Graphs using Split Decompositions. https://arxiv.org/abs/2602.23825

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