关于正接触、矩形对偶和盒正交绘图的注记
A Note on Plus-Contacts, Rectangular Duals, and Box-Orthogonal Drawings
摘要：
平面图$G$的正接触表示称为$c$ -平衡的，如果对于每个正形状$+_v$，与$+_v$的每个臂关联的其他正形状的数量至多为$ c Δ+O(1)$，其中$Δ$是$G$的最大度。虽然对于平面图的几个子类已经实现了$c$的小值（例如，$2$ -树和$3$ -树），它是未知的，是否$c$ -平衡表示与$c<1$存在于任意平面图。在本文中，我们计算$(1/2)$平衡加接触表示的所有平面图，承认一个矩形对偶。我们的结果意味着任何具有矩形对偶的图都有一个1-弯曲盒正交图，使得对于每个顶点$v$，表示$v$的盒是边长为$\frac{deg(v)}{2}+ O(1)$的正方形。
英文摘要：
A plus-contact representation of a planar graph $G$ is called $c$-balanced if for every plus shape $+_v$, the number of other plus shapes incident to each arm of $+_v$ is at most $ c Δ+O(1)$, where $Δ$ is the maximum degree of $G$. Although small values of $c$ have been achieved for a few subclasses of planar graphs (e.g., $2$- and $3$-trees), it is unknown whether $c$-balanced representations with $c<1$ exist for arbitrary planar graphs. In this paper we compute $(1/2)$-balanced plus-contact representations for all planar graphs that admit a rectangular dual. Our result implies that any graph with a rectangular dual has a 1-bend box-orthogonal drawings such that for each vertex $v$, the box representing $v$ is a square of side length $\frac{deg(v)}{2}+ O(1)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
