We prove that, given two topologically-equivalent upward planar straight-line drawings of an n -vertex directed graph G , there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of O (1) morphing steps if G is a reduced planar st -graph, O ( n ) morphing steps if G is a planar st -graph, O ( n ) morphing steps if G is a reduced upward planar graph, and $$O(n^2)$$ O ( n 2 ) morphing steps if G is a general upward planar graph. Further, we show that $$\varOmega (n)$$ Ω ( n ) morphing steps might be necessary for an upward planar morph between two topologically-equivalent upward planar straight-line drawings of an n -vertex path.

中文翻译：

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向上平面变形

我们证明，给定一个 n 顶点有向图 G 的两个拓扑等价的向上平面直线图，它们之间总是存在一个变形，使得该变形的所有中间图都是向上平面和直线。这种变形包括 O (1) 个变形步骤，如果 G 是一个简化的平面图，O ( n ) 个变形步骤，如果 G 是一个平面 st 图，O ( n ) 个变形步骤，如果 G 是一个简化的向上平面图, 和 $$O(n^2)$$ O ( n 2 ) 变形步骤，如果 G 是一般向上的平面图。此外，我们表明 $$\varOmega (n)$$ Ω ( n ) 变形步骤对于 n 顶点路径的两个拓扑等效的向上平面直线图之间的向上平面变形可能是必要的。