Characterization of expelling materialized graphs

Completely characterize the materialized graphs whose toric billiard trajectories are all noncontractible, generalizing the characterization of expelling graphs in the absence of reflection edges.

Background

A materialized graph partitions its edges into reflection edges and refraction edges, producing a billiard system with mirrors as well as metalenses. The paper gives a sufficient condition for such a materialized graph to be expelling: the refraction-edge set must be nonempty and the graph obtained by contracting all reflection edges must be bipartite. Unlike the reflection-free case, the converse fails, so a complete characterization remains unresolved.