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.

翻译：k-平面图（通常针对k值较小的情况，如1、2、3）近期受到广泛研究，其允许边交叉，但每条边至多承载k次交叉。在近期提出的[Graph Drawing 2023]图的最小k-平面绘图中，边可能承载超过k次交叉，但在任意两条交叉边中，至少有一条边的交叉次数不超过k。这两个概念均可考虑一般绘图或称为简单（有时也称"良好"）绘图的常用受限概念。在简单绘图中，任意两条边至多交叉一次，且共享顶点的任意两条边禁止交叉。因此，我们具有一般（最小）k-平面图与简单（最小）k-平面图两个不同的概念，但现有论文常未充分区分。我们证明这种区分确实需要谨慎处理，通过证明：对于每个k≥4，存在具有一般k-平面绘图但不具有简单k-平面绘制的图；对于每个k≥2，存在具有一般最小k-平面绘图但不具有简单最小k-平面绘制的图。另一方面，当k小于这些界值时，可以不失一般性地假设为简单绘图。