Triangle-Free Planar Graphs and Segment Intersection Graphs
Natalia de Castro, Francisco Javier Molina Cobos, Juan Carlos Dana, Alberto Márquez, Marc Noy · Journal of Graph Algorithms and Applications · 2002
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.