Embedding Four-directional Paths on Convex Point Sets

TL;DR

This paper investigates how to embed directed paths with specific directional constraints onto convex point sets in the plane, ensuring planarity and directionality are preserved.

Contribution

It provides a comprehensive analysis of planar embeddings for three- and four-directional paths on convex point sets, characterizing when such embeddings are possible.

Findings

Complete characterization of embeddings for four-directional paths.

Conditions for existence of planar embeddings on convex sets.

Insights into the structure of direction-consistent embeddings.

Abstract

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.