Saturated $k$-Plane Drawings with Few Edges

Abstract

A drawing of a graph is $k$-plane if no edge is crossed more than $k$ times. In this paper we study saturated $k$-plane drawings with few edges. This are $k$-plane drawings in which no edge can be added without violating $k$-planarity. For every number of vertices $n>k+1$, we present a tight construction with $\frac{n-1}{k+1}$ edges for the case in which the edges can self-intersect. If we restrict the drawings to be $\ell$-simple we show that the number of edges in saturated $k$-plane drawings must be higher. We present constructions with few edges for different values of $k$ and $\ell$. Finally, we investigate saturated straight-line $k$-plane drawings.

This article is partially superseded by and merged with arXiv:2012.08631