Local Equivalence Classes of Distance-Hereditary Graphs using Split Decompositions

2026-02-27

AI Summary

This study addresses the problem of determining the size of graph equivalence classes induced by local complementation, a question that is highly challenging for general graphs. Focusing on several classes of distance-hereditary graphs—including complete multipartite graphs, clique-star graphs, and cograph-based relay graphs—the authors derive explicit formulas for the equivalence class sizes by leveraging split decomposition, combined with symmetry analysis and combinatorial enumeration. This work overcomes the previous limitation to paths and cycles, establishing precise theoretical characterizations and demonstrating the tightness of the derived upper bounds through structural symmetries. Consequently, it significantly broadens the family of graph classes for which local equivalence class sizes are exactly solvable.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• local complement
• equivalence classes
• distance-hereditary graphs
• split decomposition
• graph enumeration

Innovation

• split decomposition
• distance-hereditary graphs
• local complementation
• equivalence classes
• combinatorial enumeration