Mirror graphs: graph theoretical characterization of reflection arrangements and finite Coxeter groups

TL;DR

This paper characterizes mirror graphs as Cayley graphs of finite Coxeter groups, linking them to reflection arrangements, and provides an efficient algorithm for their recognition.

Contribution

It offers a complete structural characterization of mirror graphs as Cayley graphs of finite Coxeter groups and introduces a polynomial-time recognition algorithm.

Findings

Mirror graphs are exactly the Cayley graphs of finite Coxeter groups.

A polynomial algorithm for recognizing mirror graphs is developed.

The characterization connects mirror graphs to well-understood geometric structures.

Abstract

Mirror graphs were introduced by Bre\v{s}ar et al. in 2004 as an intriguing class of graphs: vertex-transitive, isometrically embeddable into hypercubes, having a strong connection with regular maps and polytope structure. In this article we settle the structure of mirror graphs by characterizing them as precisely the Cayley graphs of the finite Coxeter groups or equivalently the tope graphs of reflection arrangements - well understood and classified structures. We provide a polynomial algorithm for their recognition.