Geodesic
A First Order Logic Definition of Beyond-Planar Graphs
Journal of Graph Algorithms and Applications, Special Issue on Graph Drawing Beyond Planarity , Tome 22 (2018) no. 1, pp. 51-66.

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Beyond-planarity is a collective term for classes of graphs that extend the planar graphs and are defined by drawings with restrictions on crossings. Examples are 1-planar, fan-planar, fan-crossing free, and quasi-planar graphs. We define these and other classes of beyond-planar graphs by simple first order formulas, using two predicates to express a crossing and an adjacency of two edges, and establish inclusion relationships between the so obtained graph classes. 
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     title = {A {First} {Order} {Logic} {Definition} of {Beyond-Planar} {Graphs}},
     journal = {Journal of Graph Algorithms and Applications},
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     doi = {10.7155/jgaa.00455},
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     url = {http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00455/}
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Franz Brandenburg. A First Order Logic Definition of Beyond-Planar Graphs. Journal of Graph Algorithms and Applications, 
							Special Issue on Graph Drawing Beyond Planarity
					, Tome 22 (2018) no. 1, pp. 51-66. doi : 10.7155/jgaa.00455. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00455/

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