Monotone drawings of graphs with few directions

A monotone drawing of a graph is a straight-line planar drawing such that every pair of vertices is connected by a path that monotonically increases with respect to a direction. However, different pairs of vertices might use different directions of monotonicity. In this paper we aim at constructing monotone drawings in which the number of different directions of monotonicity used by all the pairs of vertices is small. We show that a planar graph admits a monotone drawing with only one direction of monotonicity if and only if it is Hamiltonian. Also, we prove that maximal planar graphs admit monotone drawings with two (orthogonal) directions, while triconnected planar graphs with three directions. The latter two results are obtained by applying the famous drawing algorithm by Schnyder.