Recognition of simple polygon visibility graphs

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

Background

Visibility graphs encode which pairs of polygon vertices see each other with line-of-sight inside the polygon. While hardness and completeness are known for visibility graphs with holes and for external/internal variants, the fundamental recognition problem for simple polygons has resisted classification.

The authors explicitly note that even providing the ordering of vertices along the polygon does not settle the problem.