Group-theoretical graph categories

TL;DR

This paper generalizes the concept of group-theoretical categories of partitions to graph categories, replacing symmetric groups with automorphism groups of graphs, expanding the framework for analyzing symmetries in algebraic structures.

Contribution

It introduces a generalization of existing partition categories to graph categories, broadening the scope of algebraic and combinatorial symmetry analysis.

Findings

Generalization of group-theoretical categories to graph categories

Extension of semidirect product descriptions to automorphism groups of graphs

Framework applicable to a wider class of algebraic structures involving graph symmetries

Abstract

The semidirect product of a finitely generated group dual with the symmetric group can be described through so-called group-theoretical categories of partitions (covers only a special case; due to Raum--Weber, 2015) and skew categories of partitions (more general; due to Maassen, 2018). We generalize these results to the case of graph categories, which allows to replace the symmetric group by the group of automorphisms of some graph.