Embedding Four-directional Paths on Convex Point Sets

A directed path whose edges are assigned labels "up", "down", "right", or "left" is called \emph{four-directional}, and \emph{three-directional} if at most three out of the four labels are used. A \emph{direction-consistent embedding} of an \mbox{$n$-vertex} four-directional path $P$ on a set $S$ of $n$ points in the plane is a straight-line drawing of $P$ where each vertex of $P$ is mapped to a distinct point of $S$ and every edge points to the direction specified by its label. We study planar direction-consistent embeddings of three- and four-directional paths and provide a complete picture of the problem for convex point sets.