Non-stretchable pseudo-visibility graphs

We exhibit a family of graphs which can be realized as pseudo-visibility graphs of pseudo-polygons, but not of straight-line polygons. The example is based on the characterization of vertex-edge pseudo-visibility graphs of O'Rourke and Streinu [Proc. ACM Symp. Comput. Geometry, Nice, France, 1997, pp. 119-128] and extends a recent result of the author [Proc. ACM Symp. Comput. Geometry, Miami Beach, 1999, pp. 274-280] on non-stretchable vertex-edge visibility graphs. We construct a pseudo-visibility graph for which there exists a unique compatible vertex-edge visibility graph, which is then shown to be non-stretchable. The construction is then extended to an infinite family.