Geodesic
Drawing Outer 1-planar Graphs with Few Slopes
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. 707-741.

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A graph is outer 1-planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1-planar graphs are a superclass of the outerplanar graphs and a subclass of the planar partial 3-trees. We show that an outer 1-planar graph G of bounded degree ∆ admits an outer 1-planar straight-line drawing that uses O(∆) different slopes, which generalizes a previous result by Knauer et al. about the outerplanar slope number of outerplanar graphs (Knauer, Micek, and Walczak. CGTA, 2014). We also show that O(∆2) slopes suffice to construct a crossing-free straight-line drawing of G; the best known upper bound on the planar slope number of planar partial 3-trees of bounded degree ∆ is O(∆5) as proved by Jelínek et al. (V. Jelínek, E. Jelínková, J. Kratochvíl, B. Lidický, M. Tesar, and T. Vyskocil. Graphs and Combinatorics, 2013). 
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     author = {Emilio Di Giacomo and Giuseppe Liotta and Fabrizio Montecchiani},
     title = {Drawing {Outer} 1-planar {Graphs} with {Few} {Slopes}},
     journal = {Journal of Graph Algorithms and Applications},
     pages = {707--741},
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Emilio Di Giacomo; Giuseppe Liotta; Fabrizio Montecchiani. Drawing Outer 1-planar Graphs with Few Slopes. 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. 707-741. doi : 10.7155/jgaa.00376. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00376/

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