Geodesic
Triangle-Free Planar Graphs and Segment Intersection Graphs
Journal of Graph Algorithms and Applications, Selected Papers from the 1999 Symposium on Graph Drawing , Tome 6 (2002) no. 1, pp. 7-26.

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We prove that every triangle-free planar graph is the intersection graph of a set of segments in the plane. Moreover, the segments can be chosen in only three directions (horizontal, vertical and oblique) and in such a way that no two segments cross, i.e., intersect in a common interior point. This particular class of intersection graphs is also known as contact graphs. 
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Natalia de Castro; Francisco Javier Cobos; Juan Carlos Dana; Alberto Marquez; Marc Noy. Triangle-Free Planar Graphs and Segment Intersection Graphs. Journal of Graph Algorithms and Applications, 
							Selected Papers from the 1999 Symposium on Graph Drawing
					, Tome 6 (2002) no. 1, pp. 7-26. doi : 10.7155/jgaa.00043. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00043/

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