An interesting class of orthogonal representations consists of the so-called turn-regular ones, i.e., those that do not contain any pair of reflex corners that "point to each other" inside a face. For such a representation H it is possible to compute in linear time a minimum-area drawing, i.e., a drawing of minimum area over all possible assignments of vertex and bend coordinates of H. In contrast, finding a minimum-area drawing of H is NP-hard if H is non-turn-regular. This scenario naturally motivates the study of which graphs admit turn-regular orthogonal representations. In this paper we identify notable classes of biconnected planar graphs that always admit such representations, which can be computed in linear time. We also describe a linear-time testing algorithm for trees and provide a polynomial-time algorithm that tests whether a biconnected plane graph with "small" faces has a turn-regular orthogonal representation without bends.

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关于转正则正交表示

一类有趣的正交表示由所谓的常规转弯表示组成，即那些不包含任何一对在面内“相互指向”的反射角的表示。对于这样的表示 H，可以在线性时间内计算最小面积图，即在 H 的所有可能的顶点和弯曲坐标分配上的最小面积图。相反，找到 H 的最小面积图是如果 H 是非转弯规则的，则为 NP-hard。这种情况自然会激发对哪些图允许转弯正则正交表示的研究。在本文中，我们确定了一些值得注意的双连通平面图，它们总是允许这种表示，可以在线性时间内计算。