CHARACTERIZING AND RECOGNIZING THE VISIBILITY GRAPH OF A FUNNEL-SHAPED POLYGON

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 a linear time Real RAM algorithm for drawing a funnel on the plane that corresponds to an F-graph. We next give a linear-time algorithm for recognizing an F-graph. When the algorithm recognizes an F-graph, it also reports one of the Hamiltonian cycles defining the boundary of its corresponding funnel. This recognition algorithm takes linear time even on a RAM. We finally show that an F-graph is weakly triangulated and therefore perfect, which agrees with the fact that perfect graphs are related to geometric structures.