A connected topological drawing of a graph divides the plane into a number of cells. The type of a cell $c$ is the cyclic sequence of crossings and vertices along the boundary walk of $c$. For example, all triangular cells with three incident crossings and no incident vertex share the same cell type. When a non-homotopic drawing of an $n$-vertex multigraph $G$ does not contain any such cells, Ackerman and Tardos [JCTA 2007] proved that $G$ has at most $8n-20$ edges, while Kaufmann, Klemz, Knorr, Reddy, Schröder, and Ueckerdt [GD 2024] showed that this bound is tight. In this paper, we initiate the in-depth study of non-homotopic drawings that do not contain one fixed cell type \celltype, and investigate the edge density of the corresponding multigraphs, i.e., the maximum possible number of edges. We consider non-homotopic as well as simple drawings, multigraphs as well as simple graphs, and every possible type of cell. For every combination of drawing style, graph type, and cell type, we give upper and lower bounds on the corresponding edge density. With the exception of the cell type with four incident crossings and no incident vertex, we show for every cell type \celltype that the edge density of $n$-vertex (multi)graphs with \celltype-free drawings is either linear in $n$ or superlinear in $n$. In most cases, our bounds are tight up to an additive constant. We further consider cell types that are not incident to any crossing in more detail and find that all connected simple graphs but a short list of exceptions admit a simple drawing that does not contain any such cells. Additionally, we improve the current lower bound on the edge density of simple graphs that admit a non-homotopic quasiplanar drawing from $7n-28$ to $7.5n-28$.

翻译：连通拓扑图绘制将平面划分为若干单元。单元$c$的类型是沿其边界行走时遇到的交叉点和顶点的循环序列。例如，所有具有三个入射交叉点且无入射顶点的三角形单元共享相同的单元类型。当$n$顶点多重图$G$的非同伦绘制不包含任何此类单元时，Ackerman和Tardos [JCTA 2007] 证明了$G$最多有$8n-20$条边，而Kaufmann、Klemz、Knorr、Reddy、Schröder和Ueckerdt [GD 2024] 表明该界限是紧的。本文首次深入研究了不包含某一固定单元类型\celltype 的非同伦绘制，并考察了对应多重图的边密度，即可能的最大边数。我们考虑了非同伦绘制与简单绘制、多重图与简单图，以及所有可能的单元类型。针对绘制风格、图类型和单元类型的每种组合，我们给出了对应边密度的上界与下界。除了具有四个入射交叉点且无入射顶点的单元类型外，我们证明了对于每个单元类型\celltype，具有\celltype 自由绘制的$n$顶点（多重）图的边密度要么是$n$的线性函数，要么是$n$的超线性函数。在大多数情况下，我们的界限在加法常数范围内是紧的。我们进一步详细考察了不包含任何交叉点的单元类型，发现除少数例外情况外，所有连通简单图都允许不包含此类单元的简单绘制。此外，我们将允许非同伦拟平面绘制的简单图的边密度下界从$7n-28$提升至$7.5n-28$。