Recognition of Simple Polygon Visibility Graphs

Determine the computational complexity of recognizing visibility graphs of simple polygons, including the variant in which the cyclic order of vertices along the polygon boundary is given.

Background

Visibility graphs record which pairs of polygon vertices are mutually visible within the polygon, a classical structure in computational geometry with many applications.

While recognition is ER-complete for polygons with holes, the simple polygon case—and the case with prescribed boundary ordering—remains unresolved.

References

Still, the computational complexity of the recognition problem for simple polygon visibility graphs remains open, even for the variant in which the ordering of the vertices along the polygon is given.

— The Existential Theory of the Reals as a Complexity Class: A Compendium  (2407.18006 - Schaefer et al., 2024) in Section “Visibility Graphs”

Problem 10.1 (Realizing arithmetic visibility graphs). For d ≥ 1, H ∈ [1, ∞], and a real number field K, characterize the graphs Visd,H (P, T ) arising from finite target sets T ⊂ P (K) in rational simple polygons P . How does fixing the polygon or requiring the targets to be its vertices change the question?

— Arithmetic Visibility in Polygons: Curved Sight in the Art Gallery  (2610.19506 - Barrett, 14 Oct 2026) in Problem 10.1, Section 10.1