The study of nonplanar drawings of graphs with restricted crossing configurations is a well-established topic in graph drawing, often referred to as beyond-planar graph drawing. One of the most studied types of drawings in this area are the $k$-planar drawings $(k \geq 1)$, where each edge cannot cross more than $k$ times. We generalize $k$-planar drawings, by introducing the new family of min-$k$-planar drawings. In a min-$k$-planar drawing edges can cross an arbitrary number of times, but for any two crossing edges, one of the two must have no more than $k$ crossings. We prove a general upper bound on the number of edges of min-$k$-planar drawings, a finer upper bound for $k=3$, and tight upper bounds for $k=1,2$. Also, we study the inclusion relations between min-$k$-planar graphs (i.e., graphs admitting min-$k$-planar drawings) and $k$-planar graphs.

翻译：具有受限交叉配置的非平面图绘制是图绘制领域中一个成熟的研究方向，常被称为超平面图绘制。该领域最受关注的绘制类型之一是 $k$ -平面画法 $(k \geq 1)$，其中每条边交叉次数不超过 $k$ 次。我们通过引入新的极小 $k$ -平面画法家族来推广 $k$ -平面画法。在极小 $k$ -平面画法中，边可以交叉任意次数，但对于任意两条交叉边，其中一条边的交叉次数必须不超过 $k$ 次。我们证明了极小 $k$ -平面画法边数的一个一般上界、$k=3$ 时的更精细上界，以及 $k=1,2$ 时的紧上界。此外，我们还研究了极小 $k$ -平面图（即允许极小 $k$ -平面画法的图）与 $k$ -平面图之间的包含关系。