Edge-minimum saturated k -planar drawings

Abstract

For a class $D$ of drawings of loopless (multi-)graphs in the plane, a drawing $D\in D$ is saturated when the addition of any edge to $D$ results in $D^{\prime }\notin D$—this is analogous to saturated graphs in a graph class as introduced by Turán and Erdős, Hajnal, and Moon. We focus on $k$-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most $k$ times, and the classes $D$ of all $k$-planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated $k$-planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all $n$-vertex saturated $k$-planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest $n$-vertex saturated $k$-planar drawings have $\frac{2}{k-\left(k\mathrm{mod}2\right)}\left(n-1\right)$ edges for any $k\ge 4$, while if all that is forbidden, the sparsest such drawings have $\frac{2\left(k+1\right)}{k\left(k-1\right)}\left(n-1\right)$ edges for any $k\ge 6$.