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 characterize 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.

中文翻译：

---

使用给定的接触方向扩展矩形对偶的部分表示

图$ G $的矩形对偶是$ G $由轴对齐矩形的接触表示，这样（i）没有四个矩形共享一个点，并且（ii）所有矩形的并集都是一个矩形。矩形对偶的局部表示扩展问题询问是否可以将给定的部分矩形对偶扩展为矩形对偶，即是否存在其中某些顶点由指定矩形表示的矩形对偶。组合起来，矩形对角线可以用规则的边缘标记（REL）描述，矩形边缘标记（REL）确定矩形触点的方向。我们表征了允许扩展的REL，这导致了线性时间测试算法。肯定的是，我们可以构造线性时间的扩展。