Inverse monoids of partial graph automorphisms

TL;DR

This paper characterizes the algebraic structure of inverse monoids formed by partial automorphisms of finite graphs, extending the analysis to directed and edge-colored graphs using inverse semigroup theory.

Contribution

It provides a detailed algebraic description and characterization of inverse monoids of partial automorphisms, including extensions to digraphs and edge-colored digraphs.

Findings

Describes the structure of inverse monoids of partial graph automorphisms

Provides a characterization of such inverse monoids

Extends results to digraphs and edge-colored digraphs

Abstract

A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well.