Geodesic
Multi-level Verticality Optimization: Concept, Strategies, and Drawing Scheme
Journal of Graph Algorithms and Applications, Tome 17 (2013) no. 3, pp. 329-362.

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In traditional multi-level graph drawing-known as Sugiyama's framework-the number of crossings is considered one of the most important goals. Herein, we propose the alternative concept of optimizing the verticality of the drawn edges. We formally specify the problem, discuss its relative merits, and show that drawings that are good w.r.t. verticality in fact also have a low number of crossings. We present heuristic and exact approaches to tackle the verticality problem and study them in practice. Furthermore, we present a new drawing scheme (inherently bundling edges and drawing them monotonously), especially suitable for verticality optimization. It works without the traditional subdivision of edges, i.e., edges may span multiple levels, and therefore potentially allows to tackle larger graphs.
DOI : 10.7155/jgaa.00296
Keywords: graph drawing, multi-layer drawings, Sugiyama, semi-definite programming, heuristics, non-proper graphs, crossing minimization 
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Markus Chimani; Philipp Hungerländer. Multi-level Verticality Optimization: Concept, Strategies, and Drawing Scheme. Journal of Graph Algorithms and Applications, Tome 17 (2013) no. 3, pp. 329-362. doi : 10.7155/jgaa.00296. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00296/

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