具有给定切触方向的矩形对偶的部分表示的扩展
Extending Partial Representations of Rectangular Duals with Given Contact Orientations
摘要：
图$G$的矩形对偶是图$G$通过轴对齐的矩形的接触表示，使得（i）~没有四个矩形共享一个点并且（ii）~所有矩形的并集是矩形。矩形对偶的部分表示扩张问题是关于给定的部分矩形对偶是否可以扩张为矩形对偶，即是否存在一个矩形对偶，其中某些顶点由指定的矩形表示。组合地，矩形对偶可以由确定矩形接触的方向的规则边缘标记（REL）来描述。我们描述了两种方法来解决部分表示扩展问题的矩形图与给定的REL。一方面，我们对允许扩展的REL进行扩展，从而得到线性时间测试算法。在肯定的情况下，我们可以在线性时间中构造一个扩张。这个部分表示扩展问题也可以用线性规划（LP）来表示。我们使用这个LP来解决矩形图的情况下，同时表示问题时，每个输入图与REL一起给出。
英文摘要：
A rectangular dual of a graph $G$ is a contact representation of $G$ by axis-aligned rectangles such that (i)~no four rectangles share a point and (ii)~the union of all rectangles is a rectangle. The partial representation extension problem for rectangular duals asks whether a given partial rectangular dual can be extended to a rectangular dual, that is, whether there exists a rectangular dual where some vertices are represented by prescribed rectangles. Combinatorially, a rectangular dual can be described by a regular edge labeling (REL), which determines the orientations of the rectangle contacts. We describe two approaches to solve the partial representation extension problem for rectangular duals with given REL. On the one hand, we characterise the RELs that admit an extension, which leads to a linear-time testing algorithm. In the affirmative, we can construct an extension in linear time. This partial representation extension problem can also be formulated as a linear program (LP). We use this LP to solve the simultaneous representation problem for the case of rectangular duals when each input graph is given together with a REL.
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。
