Note on k-Planar and Min-k-Planar Drawings of Graphs

The k-planar graphs, which are (usually with small values of k such as 1, 2, 3) subject to recent intense research, admit a drawing in which edges are allowed to cross, but each one edge is allowed to carry at most k crossings. In recently introduced [Binucci et al., GD 2023] min-k-planar drawings of graphs, edges may possibly carry more than k crossings, but in any two crossing edges, at least one of the two must have at most k crossings. In both concepts, one may consider general drawings or a popular restricted concept of drawings called simple (sometimes also `good'). In a simple drawing, every two edges are allowed to cross at most once, and any two edges which share a vertex are forbidden to cross. While, regarding the former concept, it is for k<=3 known (but not widely known) that every general k-planar graph admits a simple k-planar drawing and this ceases to be true for any k>=4, the difference between general and simple drawings in the latter concept is more striking. We prove that graphs with a min-k-planar drawing but no simple min-k-planar drawing exist for every k>=2, and for every k>=3 there even is a graph with a min-3-planar drawing but no simple min-k-planar drawing.