Geodesic
1-Visibility Representations of 1-Planar Graphs
Journal of Graph Algorithms and Applications, Tome 18 (2014) no. 3, pp. 421-438.

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A 1-visibility representation of a graph displays each vertex as a horizontal vertex-segment, called a bar, and each edge as a vertical edge-segment between the segments of the vertices, such that each edge-segment crosses at most one vertex-segment and each vertex-segment is crossed by at most one edge-segment. A graph is 1-visible if it has such a representation. 1-visibility is related to 1-planarity where graphs are drawn such that each edge is crossed at most once, and specializes bar 1-visibility where vertex-segments can be crossed many times. We develop a linear time algorithm to compute a 1-visibility representation of an embedded 1-planar graph in O(n2) area. Hence, every 1-planar graph is 1-visible. Concerning density, both 1-visible and 1-planar graphs of size n have at most 4n−8 edges. However, for every n ≥ 7 there are 1-visible graphs with 4n−8 edges, which are not 1-planar. 
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     author = {Franz Brandenburg},
     title = {1-Visibility {Representations} of {1-Planar} {Graphs}},
     journal = {Journal of Graph Algorithms and Applications},
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     volume = {18},
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     year = {2014},
     doi = {10.7155/jgaa.00330},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00330/}
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Franz Brandenburg. 1-Visibility Representations of 1-Planar Graphs. Journal of Graph Algorithms and Applications, Tome 18 (2014) no. 3, pp. 421-438. doi : 10.7155/jgaa.00330. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00330/

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