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. In our setting we only allow simple drawings, that is, any two edges cross at most once, no two adjacent edges cross, and no three edges intersect at a common crossing point.

翻译：具有受限交叉构型的图的非平面画法研究是图绘制领域中一个成熟的方向，常被称为超平面图绘制。该领域最受关注的画法类型之一是$k$-平面画法$(k \geq 1)$，其中每条边交叉次数不超过$k$次。我们通过引入新的最小-$k$-平面画法家族来推广$k$-平面画法。在最小-$k$-平面画法中，边可以任意次交叉，但对于任意两条交叉边，其中一条的交叉次数必须不超过$k$次。我们证明了一般情况下最小-$k$-平面画法边数的上界，针对$k=3$的更精细上界，以及$k=1,2$的紧上界。此外，我们研究了最小-$k$-平面图（即允许最小-$k$-平面画法的图）与$k$-平面图之间的包含关系。在我们的设定中仅考虑简单画法，即任意两条边至多交叉一次，没有两条相邻边交叉，且没有三条边交于同一交叉点。