Complete characterization of ensnaring graphs

Characterize all ensnaring graphs, thereby determining precisely which finite graphs have all toric combinatorial refraction billiard trajectories contractible.

Background

The paper establishes several necessary and sufficient conditions for a graph to be ensnaring, including classifications for bipartite graphs, complete graphs, cycles, certain wedges, and complements of selected graph families. However, these results do not provide a complete classification of ensnaring graphs. The authors identify such a characterization as a broad unresolved problem and then propose more specific conjectures that may yield structural information.

References

There are still many questions concerning ensnaring graphs that we have not answered. Of course, it would be wonderful to have a complete characterization of ensnaring graphs, though that seems quite difficult at the moment.

— Homology in Combinatorial Refraction Billiards  (2502.06013 - Defant et al., 9 Feb 2025) in Section 6, Future Directions, subsection Ensnaring Graphs