Min-k-planar Drawings of Graphs

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 ≥ 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 point.