Every Schnyder drawing is a greedy embedding

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In this paper, we show that every Schnyder drawing is a greedy embedding. Schnyder drawings are used to represent planar (maximal) graphs. It is a way of getting coordinates in R2 given a graph G=(V,E) such that the representation is planar. The Schnyder technique leads to a family of representations and previous results show that a particular representation may be chosen such that the drawing has additional properties like being greedy or monotone. In this article, we relax the definition of greediness to a definition that does not rely on the geometry and the Euclidean distance in R2, but rather on the combinatorial graph G. The construction of greedy paths valid for all Schnyder representations shows that, provided the relaxed definition, every Schnyder drawing is a greedy embedding.