On RAC Drawings of Graphs with one Bend per Edge

08/30/2018

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by   Patrizio Angelini, et al.

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A k-bend right-angle-crossing drawing or (k-bend RAC drawing, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are 90 degrees. Accordingly, a graph that admits a k-bend RAC drawing is referred to as k-bend right-angle-crossing graph (or k-bend RAC, for short). In this paper, we continue the study of the maximum edge-density of 1-bend RAC graphs. We show that an n-vertex 1-bend RAC graph cannot have more than 5.5n-O(1) edges. We also demonstrate that there exist infinitely many n-vertex 1-bend RAC graphs with exactly 5n-O(1) edges. Our results improve both the previously known best upper bound of 6.5n-O(1) edges and the corresponding lower bound of 4.5n-O(√(n)) edges by Arikushi et al. (Comput. Geom. 45(4), 169--177 (2012)).