I've recently come across the following lemma. 


Lemma (Valiant): A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at integer coordinates and its edges are drawn so that they are made up of line segments of the form $x=i$ or $y=j$, for integers $i$ and $j$.


I'd like to first point out that i'm not interested in the size of the area that the graph is embedded in, so we can for all intents and purposes assume it's possible to work in an area of unrestricted size. I'm interested to know (because they haven't explicitly stated) whether any graph in the class mentioned, can be embedded without having any 'bends' in the edges? If it's necessary to allow for 'bends' is there a restriction on the maximum number required? Secondly would i be able to create an embedding where the lengths of all the edges are multiples of $4$ (perhaps by applying some kind stretch to the original graph)? 



Embedding of planar graphs
Bends are necessary, we have studied this problem in this paper: <http://arxiv.org/abs/1009.1315>.


