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DOI: 10.7155/jgaa.00547
Drawing Subcubic 1Planar Graphs with Few Bends, Few Slopes, and Large Angles
Vol. 25, no. 1, pp. 128, 2021. Regular paper.
Abstract We show that the $1$planar slope number of $3$connected cubic $1$planar graphs is at most four when edges are drawn as polygonal curves with at most one bend each, that is, any such graph admits a drawing with at most one bend per edge and such that the number of distinct slopes used by the edge segments is at most four.
This bound is obtained by drawings whose angular and crossing resolution is at least $\pi/4$.
On the other hand, if the embedding is fixed, then there is a $3$connected cubic $1$planar graph
that needs three slopes when drawn with at most one bend per edge.
We also show that two slopes always suffice for $1$planar drawings of subcubic $1$planar graphs with at most two bends per edge. This bound is obtained with angular resolution $\pi/2$ and the drawing has crossing resolution $\pi/2$ (i.e., it is a RAC drawing). Finally, we prove lower bounds for the slope number of straightline $1$planar drawings in terms of
number of vertices and maximum degree.
This work is licensed under the terms of the CCBY license.

Submitted: July 2019.
Reviewed: January 2020.
Revised: April 2020.
Reviewed: June 2020.
Revised: August 2020.
Accepted: November 2020.
Final: November 2020.
Published: January 2021.
Communicated by
Ignaz Rutter

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