A combinatorial view of visibility graphs of simple polygons

Abstract

This 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.

On recognizing and characterizing visibility graphs of simple polygons

“…Another counterexample ( Fig. 9) to our conjecture of sufficiency in [ 12] was proposed by Abello et al [4]. In fact this counterexample shows that the three necessary conditions are not sufficient even after strengthening Necessary Condition 3 as suggested by Everett.…”

“…Moreover, there is a unique assignment of blocking vertices to all minimal invisible pairs. The blocking vertex 1 is assigned to (2,9) and (2,10), the blocking vertex 2 is assigned to (1,3) and (1,4), the blocking vertex 5 is assigned to (1,6), (2,6), (3,6), (4,6), (4,7), and (4,8), and the blocking vertex 8 is assigned to (1, 7), (2, 7), (5,9), (6,9), (7,9), and (7,10). Observe that no blocking vertex is assigned to two or more of its separable invisible pairs.…”

“…She also suggested the stronger version of the third necessary condition, which has been proved by Srinivasaraghavan and Mukhopadhyay [21 ]. However, the counterexampie given by Abello et al [4] shows that even with the stronger version of the third necessary condition, the three necessary conditions are not sufficient. In this version we identify another necessary condition which also circumvents this counterexample.…”

A General Notion of Visibility Graphs

“…(Note that if one such line intersects C before B, all such lines that intersect C must do so before B, as B and C are convex.) Since aϕ (1) is occluded, at least one such interloping region exists. Pick the region C with t C minimal, so that aϕ(t) is unobstructed for t < t C ; if more than one such region exists, pick the one closest to a along the line aϕ(t C ).…”

Visibility Graphs and Oriented Matroids

“…Further necessary conditions were developed by Coullard and Lubiw [9], but they also showed that they are not sufficient. Abello et al [4] have strengthened these results by showing that the proposed conditions are not sufficient, even for triconnected graphs, and in the case of the conditions of [10], even for planar graphs.…”