Geodesic
Bar 1-Visibility Graphs and their relation to other Nearly Planar Graphs
Journal of Graph Algorithms and Applications, Tome 18 (2014) no. 5, pp. 721-739.

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A graph is called a strong (resp. weak) bar 1-visibility graph if its vertices can be represented as horizontal segments (bars) in the plane so that its edges are all (resp. a subset of) the pairs of vertices whose bars have a ε-thick vertical line connecting them that intersects at most one other bar. We explore the relation among weak (resp. strong) bar 1-visibility graphs and other nearly planar graph classes. In particular, we study their relation to 1-planar graphs, which have a drawing with at most one crossing per edge; quasi-planar graphs, which have a drawing with no three mutually crossing edges; and the squares of planar 1-flow networks, which are upward digraphs with in- or out-degree at most one. Our main results are that 1-planar graphs and the (undirected) squares of planar 1-flow networks are weak bar 1-visibility graphs and that these are quasi-planar graphs. 
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     author = {William Evans and Michael Kaufmann and William Lenhart and Tamara Mchedlidze and Stephen Wismath},
     title = {Bar {1-Visibility} {Graphs} and their relation to other {Nearly} {Planar} {Graphs}},
     journal = {Journal of Graph Algorithms and Applications},
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William Evans; Michael Kaufmann; William Lenhart; Tamara Mchedlidze; Stephen Wismath. Bar 1-Visibility Graphs and their relation to other Nearly Planar Graphs. Journal of Graph Algorithms and Applications, Tome 18 (2014) no. 5, pp. 721-739. doi : 10.7155/jgaa.00343. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00343/

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