In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes as input an upward planar drawing ${\mathrm{\Gamma }}_H$ of a subgraph H of a directed graph G and asks whether ${\mathrm{\Gamma }}_H$ can be extended to an upward planar drawing of G. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing.

We show the following results.

–

First, we prove that the Upward Planarity Extension problem is NP-complete, even if G has a prescribed upward embedding, the vertex set of H coincides with the one of G, and H contains no edge.

–

Second, we show that the Upward Planarity Extension problem can be solved in $O(n\log n)$ time if G is an n-vertex upward planar st-graph. This result improves upon a known $O(n^2)$-time algorithm, which however applies to all n-vertex single-source upward planar graphs.

–

Finally, we show how to solve in polynomial time a surprisingly difficult version of the Upward Planarity Extension problem, in which the underlying graph of G is a path or a cycle, G has a prescribed upward embedding, H contains no edges, and no two vertices share the same y-coordinate in ${\mathrm{\Gamma }}_H$.

在本文中，我们研究了向上平面扩展问题的计算复杂度，该问题以向上平面图作为输入${\mathrm{\Gamma }}_H$有向图G的子图H的问${\mathrm{\Gamma }}_H$可以扩展到G的向上平面图。我们的研究适合于部分表示的可扩展性研究，最近它已成为Graph Drawing中的主流。

我们显示以下结果。

–

首先，我们证明了向上平面扩展问题是NP完全的，即使G具有规定的向上嵌入，H的顶点集与G的顶点集重合，并且H不包含边。

–

其次，我们证明了向上的平面性扩展的问题可以得到解决$Ø（ñ\mathrm{日志}ñ）$时间如果ģ是ñ -点向上平面ST -图。这个结果比已知的要好$Ø（{ñ}^2）$时间算法，但是它适用于所有n个顶点单源向上平面图。

–

最后，我们展示了如何在多项式时间内求解令人惊讶的版本的“向上平面扩展”问题，其中G的基础图是路径或循环，G具有规定的向上嵌入，H不包含边，并且不包含两个顶点共享相同的y坐标${\mathrm{\Gamma }}_H$。