For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the plane, a drawing $D \in \mathcal{D}$ is \emph{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. 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 $k \geq 6$.

翻译：对于平面上无线平面( muldial) 平面( muldial) 平面( muldial) 的绘图 $2, mathcal {D) $, 当将任何边缘加到$D 美元导致$D\ notin\ mathcal{D} $ - 这类似于图类中饱和的图表, 由 Tur\ an (1941) 和 Erd\ H{o} 、 Hajnal 和 Moon (1964) 引入。 我们关注的是 $k 的平面图, 也就是在每个边缘超过 $k 美元时绘制的图表 。 $\ mathcal{D}, 所有 $k 平面的平面图都符合一些限制, 例如没有跨过任何事件边缘, 没有一对边缘超过一次, 或没有边缘跨过6 。 虽然饱和 $ $ 美元 平面图是前几个工程的焦点, 在平面平面的平面图上如何稀少的平面上,, 美元 。 当所有平面平面平面平面平面平面 4, 我们建立一个普通的平面框架 。