Point Visibility Graphs and O-Convex Cover

A visibility relation can be viewed as a graph: the uncountable graph of a visibility relationship between points in a polygon P is called the point visibility graph (PVG) of P. In this paper we explore the use of perfect graphs to characterize tractable subprob-lems of visibility problems. Our main result is a characterization of which polygons are guaranteed to have weakly triangulated PVGs, under a generalized notion of visibility called O-visibility. Let O denote a set of line orientations. A connected point set P is called O-convex if the intersection of P with any line with orientation in O is connected. Two points in a polygon are said to be O-visible if there is an O-convex path between them inside the polygon. Let O ? denote the set of orientations perpendicular to orientations in O. Let O 0 O ? be the set of orientations such that a \reeex" local maximum in the boundary of P exists with respect to. Our characterization of which polygons have weakly-triangulated PVGs is based on restricting the cardinality and span of O 0. This characterization allows us to exhibit a class of polygons admitting a polynomial algorithm for O-convex cover. We also show that for any nite cardinality O, O-convex cover and O-star cover are in NP, and have polynomial time algorithms for any xed covering number. Our results imply previous results for the special case of O = f0; 90 g of Culberson and Reckhow,