Geodesic
Embedding Four-directional Paths on Convex Point Sets
Journal of Graph Algorithms and Applications, Special Issue on Selected Papers from the Twenty-second International Symposium on Graph Drawing, GD 2014 , Tome 19 (2015) no. 2, pp. 743-759.

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A directed path whose edges are assigned labels "up", "down", "right", or "left" is called four-directional, and three-directional if at most three out of the four labels are used. A direction-consistent embedding of an n-vertex three- or 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. 
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     author = {Oswin Aichholzer and Thomas Hackl and Sarah Lutteropp and Tamara Mchedlidze and Birgit Vogtenhuber},
     title = {Embedding {Four-directional} {Paths} on {Convex} {Point} {Sets}},
     journal = {Journal of Graph Algorithms and Applications},
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Oswin Aichholzer; Thomas Hackl; Sarah Lutteropp; Tamara Mchedlidze; Birgit Vogtenhuber. Embedding Four-directional Paths on Convex Point Sets. Journal of Graph Algorithms and Applications, 
							Special Issue on Selected Papers from the Twenty-second International Symposium on Graph Drawing, GD 2014
					, Tome 19 (2015) no. 2, pp. 743-759. doi : 10.7155/jgaa.00368. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00368/

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