Geodesic
Planar Octilinear Drawings with One Bend Per Edge
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. 657-680.

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In octilinear drawings of planar graphs, every edge is drawn as a sequence of horizontal, vertical and diagonal (45°) line segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A k-planar graph is a planar graph in which each vertex has degree at most k. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size O(n2) ×O(n). For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge for some edges.
DOI : 10.7155/jgaa.00369
Keywords: Octilinear graph drawing, Bew bends, Bounded degree graphs 
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     author = {Michael Bekos and Martin Gronemann and Michael Kaufmann and Robert Krug},
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Michael Bekos; Martin Gronemann; Michael Kaufmann; Robert Krug. Planar Octilinear Drawings with One Bend Per Edge. 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. 657-680. doi : 10.7155/jgaa.00369. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00369/

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