We consider upward-planar layered drawings of directed graphs, i.e., crossing-free drawings in which each edge is drawn as a y-monotone curve going upward from its tail to its head, and the y-coordinates of the vertices are integers. The span of an edge in such a drawing is the absolute difference between the y-coordinates of its endpoints, and the span of the drawing is the maximum span of any edge. The span of an upward-planar graph is the minimum span over all its upward-planar drawings. We study the problem of determining the span of upward-planar graphs and provide both combinatorial and algorithmic results. On the combinatorial side, we present upper and lower bounds for the span of directed trees. On the algorithmic side, we show that the problem of determining the span of an upward-planar graph is NP-complete already for directed trees and for biconnected single-source graphs. Moreover, we give efficient algorithms for several graph families with a bounded number of sources, including st-planar graphs and graphs where the planar or upward-planar embedding is prescribed. Furthermore, we show that the problem is fixed-parameter tractable with respect to the vertex cover number and the treedepth plus the span.

翻译：我们考虑有向图的向上平面分层绘制，即无交叉的绘制方式，其中每条边被绘制为从尾到头沿y方向单调上升的曲线，且顶点的y坐标为整数。在此类绘制中，边的跨度是其两端点y坐标之差的绝对值，绘制的跨度是所有边中的最大跨度。向上平面图的最小跨度定义为所有其向上平面绘制中的最小跨度值。我们研究确定向上平面图跨度的问题，并给出组合与算法两方面的结果。在组合方面，我们提出了有向树跨度的上界和下界。在算法方面，我们证明确定向上平面图跨度的问题即使对有向树和双连通单源图也是NP完全的。此外，我们为若干具有有界源数量的图族给出了高效算法，包括st平面图以及平面嵌入或向上平面嵌入已指定的图。进一步地，我们证明该问题关于顶点覆盖数以及树深度加跨度是固定参数可解的。