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Simple Topological Drawings of $k$-Planar Graphs
arXiv - CS - Computational Geometry Pub Date : 2020-08-25 , DOI: arxiv-2008.10794
Michael Hoffmann and Chih-Hung Liu and Meghana M. Reddy and Csaba D. T\'oth

Every finite graph admits a \emph{simple (topological) drawing}, that is, a drawing where every pair of edges intersects in at most one point. However, in combination with other restrictions simple drawings do not universally exist. For instance, \emph{$k$-planar graphs} are those graphs that can be drawn so that every edge has at most $k$ crossings (i.e., they admit a \emph{$k$-plane drawing}). It is known that for $k\le 3$, every $k$-planar graph admits a $k$-plane simple drawing. But for $k\ge 4$, there exist $k$-planar graphs that do not admit a $k$-plane simple drawing. Answering a question by Schaefer, we show that there exists a function $f : \mathbb{N}\rightarrow\mathbb{N}$ such that every $k$-planar graph admits an $f(k)$-plane simple drawing, for all $k\in\mathbb{N}$. Note that the function $f$ depends on $k$ only and is independent of the size of the graph. Furthermore, we develop an algorithm to show that every $4$-planar graph admits an $8$-plane simple drawing.

中文翻译:

$k$-平面图的简单拓扑图

每个有限图都允许\emph{简单(拓扑)绘图},即每对边最多在一个点相交的绘图。然而,结合其他限制,简单的图纸并不普遍存在。例如,\emph{$k$-planar graphs} 是那些可以绘制成每条边最多有 $k$ 个交叉点的图(即,它们承认 \emph{$k$-plane 绘图})。众所周知,对于$k\le 3$,每个$k$-平面图都承认一个$k$-平面简单绘图。但是对于$k\ge 4$,存在不接受$k$-平面简单绘图的$k$-平面图。回答 Schaefer 的一个问题,我们证明存在一个函数 $f : \mathbb{N}\rightarrow\mathbb{N}$ 使得每个 $k$-平面图都承认 $f(k)$-平面简单绘图,对于所有 $k\in\mathbb{N}$。请注意,函数 $f$ 仅取决于 $k$ 并且与图的大小无关。此外,我们开发了一种算法来证明每个 $4$-平面图都允许一个 $8$-平面简单绘图。
更新日期:2020-08-26
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