In a previous paper, the authors reduced the characterization of visibility graphs of simple polygons to the coordinatization of certain uniform rank 3 oriented matroids. We show here that in the case of 2-spiral polygons such coordinatization is possible therefore characterizing visibility graphs of 2-spiral polygons.

This paper describes a new set of necessary conditions for a given graph to be the visibility graph of a simple polygon. For every graph satisfying these conditions we show that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely co- ordinatizable would yield a simple polygon whose visibility graph is isomorphic to the given graph. This will in turn offer the first characterization of this class of graphs.

The recognition problem for visibility graphs of simple polygons is not known to be in NP, nor is it known to be NP-hard. It is, however, known to be inPSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans.

This dissertation is concerned with the combinatorics of visibility graphs of simple polygons. We seek to obtain combinatorial properties that characterize such visibility graphs. A key algorithmic problem that arises in this context is the reconstruction problem: given a graph as input, construct a simple polygon in the plane, such that the visibility graph of the polygon is isomorphic to the given graph, or halt within a finite amount of time if no such polygon exists. The decision problem for visibility graph recognition is only known to be in PSPACE. Algorithms for the reconstruction problem are known only for a few restricted classes of graphs. We study the relationship between the visibility graph recognition/reconstruction problem and two classical problems in computational synthetic geometry: the co-ordinatizability problem for chirotopes and the realizability problem for circular sequences of permutations. We introduce a class of graphs called Q-Persistent graphs, and show that all visibility graphs of simple polygons are properly contained in this class. We derive additional restrictions that are necessary for a graph in this class to be a visibility graph. We show that for any graph satisfying these necessary conditions, a simplicial chirotope can be constructed by a polynomial time algorithm. The existence of affine co-ordinatizations for these chirotope would imply that every graph in the corresponding class is a visibility graph, thus characterizing completely, the visibility graphs of simple polygons. Our algorithm may be viewed as solving a combinatorial version of the reconstruction problem for visibility graphs of simple polygons. For the special case of visibility graphs of convex fans, we identify a special subclass called persistent graphs and we develop algorithms that reconstruct the circular sequence of permutations associated with a configuration representing the polygon. Specifically, we show that for any persistent graph, there exists a maximal chain in the weak Bruhat order of the symmetric group associated with the graph. This maximal chain can be recovered from the graph by a polynomial time algorithm. The realizability of the circular sequence associated with this chain, would imply that every persistent graph is the visibility of a convex fan.

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