On the number of directions in visibility representations of graphs

Abstract

We consider visibility representations of graphs in which the vertices are represented by a collection O of non-overlapping convex regions on the plane. Two points x and y are visible if the straight-line segment xy is not obstructed by any object. Two objects A, B ∈ O are called visible if there exist points x ∈ A, y ∈ B such that x is visible from y. We consider visibility only for a finite set of directions. In such a representation, the given graph is decomposed into a union of unidirectional visibility graphs, for the chosen set of directions. This raises the problem of studying the number of directions needed to represent a given graph. We study this number of directions as a graph parameter and obtain sharp upper and lower bounds for the represent ability of arbitrary graphs. 1980 Mathematics Subject Classification: 68R10, 68U05 CR Categories: F.2.2.