Pseudo-Triangle Visibility Graph: Characterization and Reconstruction

Abstract

The visibility graph of a simple polygon represents visibility relations between its vertices. Knowing the correct order of the vertices around the boundary of a polygon and its visibility graph, it is an open problem to locate the vertices in a plane such that it will be consistent with this visibility graph. This problem has been solved for special cases when we know that the target is a tower, a spiral, or an anchor polygon. Knowing that a given visibility graph belongs to a simple polygon with at most three concave chains on its boundary, a pseudo-triangle, we propose a linear-time algorithm for reconstructing one of its corresponding polygons. Moreover, we introduce a set of necessary and sufficient properties for characterizing visibility graphs of pseudo-triangles and propose polynomial algorithms for checking these properties.