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.

### Right Angle Crossing Graphs and 1-planarity

#### Abstract

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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2012
9783642258770
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11391/912397`
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