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

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.

💡 Research Summary

The paper provides a complete structural characterization of “mirror graphs”, a class introduced by Brešar, Klazar, Lipovec and Mohar in 2004. Mirror graphs are defined by three intertwined properties: (i) vertex‑transitivity, (ii) an isometric embedding into a hypercube (i.e., they are partial cubes), and (iii) the existence of a “mirror partition” of the edge set. A mirror partition consists of edge‑classes (E_1,\dots,E_k) such that for each class there is an automorphism (\alpha_i) swapping the two endpoints of every edge in the class and mapping the two connected components of the graph after removal of that class onto each other.

The authors first recast the definition in the language of partial‑cube theory. The Θ‑relation on edges, defined by the distance equality (d(a,x)+d(b,y)=d(a,y)+d(b,x)), partitions the edges of any partial cube into Θ‑classes, each of which corresponds to a coordinate direction in the hypercube embedding. In a mirror graph these Θ‑classes are exactly the mirror classes.

A key early result (Lemma 2.1) shows that every mirror graph is “harmonic‑even”: each vertex has a unique antipode at distance equal to the isometric dimension (i(G)), and antipodes of adjacent vertices are adjacent as well. This property guarantees a strong form of symmetry and will be used repeatedly to control automorphisms.

The paper then proves that any two Θ‑classes are linked by a convex cycle (Lemma 2.2). A convex cycle is a subgraph in which every shortest path between its vertices stays inside the cycle; such cycles contain edges from both classes and intersect each Θ‑class an even number of times. The existence of convex cycles implies that the action of any automorphism on one Θ‑class determines its action on all others.

Lemma 2.3 and Corollary 2.4 establish the uniqueness of the “mirror automorphism” associated with a Θ‑class: if an automorphism fixes every vertex of a class, it must be the identity, and consequently each class admits at most one involutory automorphism that swaps the two sides of the corresponding cut. This yields a well‑defined map (\alpha_{xy}) for every edge (xy) in a mirror graph.

With these combinatorial tools the authors connect mirror graphs to reflection arrangements. A (realizable) reflection arrangement consists of a finite set of hyperplanes through the origin in (\mathbb{R}^n) such that the orthogonal reflection across each hyperplane permutes the whole set. The chambers (connected components of the complement) form the vertices of the tope graph; two chambers are adjacent when they are separated by a single hyperplane. It is known that the tope graph of a reflection arrangement is a partial cube, and that finite Coxeter groups are precisely the groups generated by such reflections. The Cayley graph of a finite Coxeter group with respect to its set of reflections coincides with the tope graph of the associated arrangement.

The main theorem (Theorem 2.8) proves the equivalence of three families:

1. Mirror graphs,
2. Cayley graphs of finite Coxeter groups,
3. Tope graphs of reflection arrangements (i.e., realizable oriented matroids of rank n).

Thus every mirror graph is exactly a finite Coxeter group’s Cayley graph, and conversely every such Cayley graph possesses a mirror partition and the associated mirror automorphisms. This result not only classifies mirror graphs (they fall into the four infinite families and six exceptional types of finite Coxeter groups) but also provides a graph‑theoretic characterization of reflection arrangements, a problem that is open in the broader oriented‑matroid setting.

Beyond the theoretical classification, the paper contributes an explicit polynomial‑time recognition algorithm (Algorithm 1). The algorithm proceeds as follows:

1. Compute the Θ‑classes of the input graph and test whether it is a partial cube (O(n²) time).
2. For each Θ‑class, locate convex cycles that intersect it (using known O(m n²) procedures) to infer the candidate mirror automorphism as a permutation of hypercube coordinates.
3. Verify that each candidate indeed defines a graph automorphism; if all succeed, the Θ‑classes together with their automorphisms constitute a mirror partition, confirming that the graph is a mirror graph.

The overall complexity is polynomial, specifically O(m n²), making the recognition feasible for moderately sized graphs.

In summary, the paper bridges three previously separate domains—partial‑cube graph theory, finite Coxeter group theory, and the geometry of reflection arrangements—by showing that they all describe the same combinatorial objects. It resolves the structural question about mirror graphs, supplies a complete classification, and delivers a practical algorithm for their detection, thereby advancing both the theoretical understanding and the algorithmic toolbox for researchers working at the intersection of algebraic combinatorics, discrete geometry, and graph theory.