Geodesic
Parameters of Bar k-Visibility Graphs
Journal of Graph Algorithms and Applications, Special Issue on Selected Papers from the Fourteenth International Symposium on Graph Drawing, GD 2006 , Tome 12 (2008) no. 1, pp. 5-27.

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Bar k-visibility graphs are graphs admitting a representation in which the vertices correspond to horizontal line segments, called bars, and the edges correspond to vertical lines of sight which can traverse up to k bars. These graphs were introduced by Dean et al. [] who conjectured that bar 1-visibility graphs have thickness at most 2. We construct a bar 1-visibility graph having thickness 3, disproving their conjecture. Furthermore, we define semi bar k-visibility graphs, a subclass of bar k-visibility graphs, and show tight results for a number of graph parameters including chromatic number, maximum number of edges and connectivity. Then we present an algorithm partitioning the edges of a semi bar 1-visibility graph into two plane graphs, showing that for this subclass the (geometric) thickness is indeed bounded by 2. 
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Stefan Felsner; Mareike Massow. Parameters of Bar k-Visibility Graphs. Journal of Graph Algorithms and Applications, 
							Special Issue on Selected Papers from the Fourteenth International Symposium on Graph Drawing, GD 2006
					, Tome 12 (2008) no. 1, pp. 5-27. doi : 10.7155/jgaa.00157. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00157/

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