CHARACTERIZING AND RECOGNIZING VISIBILITY GRAPHS OF FUNNEL-SHAPED POLYGONS

Abstract

A funnel, which is notable for its fundamental role in visibility algorithms, is defined as a polygon that has exactly three convex vertices two of which are connected by a boundary edge. In this paper, we investigate the visibility graph of a funnel which we call an F-graph. We first present two characterizations of an F-graph, one of whose sufficiency proof itself is an algorithm to draw a corresponding funnel on the plane in O(e) time, where e is the number of the edges in an input graph. We next give an O(e) time algorithm for recognizing an F-graph. When the algorithm recognizes graph to be an F-graph, it also reports one of the Hamiltonian cycles defining the boundary of a corresponding funnel. We finally show that an F-graph is weakly triangulated and therefore perfect. This agrees with the fact that many of perfect graphs are related to geometric structures.