Local Equivalence Classes of Distance-Hereditary Graphs using Split Decompositions

TL;DR

This paper studies local complement equivalence classes in distance-hereditary graphs, deriving explicit formulas for their sizes using split decomposition, thus advancing understanding of graph transformations and classifications.

Contribution

It extends known results by providing explicit formulas for equivalence class sizes in broad families of distance-hereditary graphs using split decomposition techniques.

Findings

Derived explicit formulas for equivalence class sizes of certain distance-hereditary graphs.

Established tight upper bounds on class sizes through combinatorial enumeration.

Applied split decomposition to analyze graph equivalence classes.

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.