A combinatorial view of visibility graphs of simple polygons

The paper describes a purely combinatorial approach to the study of visibility graphs of simple polygons. We introduce a class of polygons called generalized staircase polygons and show that the visibility graph of any simple nondegenerate polygon is isomorphic to the visibility graph of some simple generalized staircase polygon. A class of graphs called persistent graphs is defined and it is shown that the visibility graph of any simple polygon can be decomposed into a sequence of persistent graphs in a natural manner. A geometric characterization of persistent graphs is used to obtain a simple characterization of the visibility graphs of convex fans. This yields a polynomial time algorithm for the reconstruction problem for convex fans with a prescribed hamiltonian cycle