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 [Graph Drawing 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. We thus have two distinct concepts of general (min-) k-planar graphs and of simply (min-) k-planar graphs, which are often not sufficiently clearly distinguished in papers. We show that this distinction indeed has to be treated carefully, by proving that there exist graphs with a general k-planar drawing but no simple k-planar drawing for every k>=4, and graphs with a general min-k-planar drawing but no simple min-k-planar drawing for every k>=2. On the other hand, for values of k smaller than these bounds, one may assume a simple drawing without loss of generality.