Drawing Subcubic 1-Planar Graphs with Few Bends, Few Slopes, and Large Angles

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 straight-line $1$-planar drawings in terms of number of vertices and maximum degree.