Right Angle Crossing Graphs and 1-Planarity
Eades, Peter and Liotta, Giuseppe (2012) Right Angle Crossing Graphs and 1-Planarity. In: Graph Drawing 19th International Symposium, GD 2011, September 21-23, 2011, Eindhoven, The Netherlands , pp. 148-153 (Official URL: http://dx.doi.org/10.1007/978-3-642-25878-7_15).
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A Right Angle Crossing Graph (also called RAC graph for short) is a graph that has a straight-line drawing where any two crossing edges are orthogonal to each other. A 1-planar graph is a graph that has a drawing where every edge is crossed at most once. We study the relationship between RAC graphs and 1-planar graphs in the extremal case that the RAC graphs have as many edges as possible. It is known that a maximally dense RAC graph with n > 3 vertices has 4n − 10 edges. We show that every maximally dense RAC graph is 1-planar. Also, we show that for every integer i such that i ≥ 0, there exists a 1-planar graph with n = 8 + 4i vertices and 4n − 10 edges that is not a RAC graph.
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