  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.
