We propose efficient algorithms for enumerating the notorious combinatorial structures of maximal planar graphs, called canonical orderings and Schnyder woods, and the related classical graph drawings by de Fraysseix, Pach, and Pollack [Combinatorica, 1990] and by Schnyder [SODA, 1990], called canonical drawings and Schnyder drawings, respectively. To this aim (i) we devise an algorithm for enumerating special $e$-bipolar orientations of maximal planar graphs, called canonical orientations; (ii) we establish bijections between canonical orientations and canonical drawings, and between canonical orientations and Schnyder drawings; and (iii) we exploit the known correspondence between canonical orientations and canonical orderings, and the known bijection between canonical orientations and Schnyder woods. All our enumeration algorithms have $O(n)$ setup time, space usage, and delay between any two consecutively listed outputs, for an $n$-vertex maximal planar graph.

翻译：我们提出了高效枚举最大平面图中著名组合结构（称为典范序与Schnyder树）以及相关经典图绘制（即由de Fraysseix、Pach和Pollack [Combinatorica, 1990]提出的典范绘制与由Schnyder [SODA, 1990]提出的Schnyder绘制）的算法。为此，(i) 我们设计了一种枚举最大平面图特殊$e$-双极定向（称为典范定向）的算法；(ii) 建立了典范定向与典范绘制之间、以及典范定向与Schnyder绘制之间的双射关系；(iii) 我们利用了典范定向与典范序之间已知的对应关系，以及典范定向与Schnyder树之间已知的双射关系。对于具有$n$个顶点的最大平面图，我们所有枚举算法的设置时间、空间使用量以及连续列出两个输出之间的延迟均为$O(n)$。