Edge-Minimum Saturated k-Planar Drawings

For a class $\mathcal{D}$ of drawings of loopless multigraphs in the plane, a drawing $D \in \mathcal{D}$ is saturated when the addition of any edge to $D$ results in $D' \notin \mathcal{D}$. This is analogous to saturated graphs in a graph class as introduced by Tur\'an (1941) and Erd\H{o}s, Hajnal, and Moon (1964). 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 $\mathcal{D}$ 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. For $k \geq 4$, 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 $\frac{2}{k - (k \bmod 2)} (n-1)$ edges for any $k \geq 4$, while if all that is forbidden, the sparsest such drawings have $\frac{2(k+1)}{k(k-1)}(n-1)$ edges for any $k \geq 7$.