Monotone Paths in Planar Convex Subdivisions and Polytopes

Abstract

Consider a connected subdivision of the plane into n convex faces where every vertex is incident to at most edges. Then, starting from every vertex there is a path with at least (log n) edges that is monotone in some direction. This bound is the best possible. Consider now a connected subdivision of the plane into n convex faces where exactly k faces are unbounded. Then, there is a path with at least (log( n=k)= log log(n=k)) edges that is monotone in some direction. This bound is also the best possible. Our methods are constructive and lead to ecient algorithms for computing monotone paths of lengths specied above. In 3-space, we show that for every n 4, there exists a polytope P with n vertices, bounded vertex degrees, and triangular faces such that every monotone path on the 1-skeleton of P has at most O(log 2 n) edges. We also construct a polytope Q with n vertices, and triangular faces, (with unbounded degree however), such that every monotone path on the 1-skeleton of Q has at most O(logn) edges.