一种用于扩展向上平面图绘制的固定参数算法
A Fixed-Parameter Algorithm for Extending Upward Planar Drawings
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中文总结 AI 辅助
研究有向无环图向上平面图绘制的扩展问题，提出以顶点+边删除距离为参数的固定参数算法，克服以往算法局限，通过对缺失边规范集刻画，实现对有许多缺失边实例的可处理性。
中文摘要 AI 辅助
有向无环图的向上平面图绘制是指每条边都从尾部向上指向头部的平面图绘制。向上平面图绘制是有向图最自然的绘制样式之一，最近在包括绘制扩展等各种不同场景下得到了研究。在绘制扩展场景中，问题是：给定一个图\(G\)以及\(G\)的一个（通常是连通的）带有绘制\(\Gamma(H)\)的子图\(H\)，能否将\(\Gamma(H)\)完成到\(G\)的绘制？针对众多绘制样式研究了绘制扩展问题；其中绝大多数是NP难的，一种典型的规避其难处理性的方法是设计参数化算法，参数衡量预绘制图\(H\)中还缺少\(G\)的“多少”部分。本文提出一种用于扩展向上平面图绘制的固定参数算法，通过使用顶点+边删除距离作为参数克服了这一缺点，即使对于有许多缺失边的实例也能实现可处理性。我们结果的一个关键要素是对在绘制中与水平线段交叉的缺失边的“规范”集的新颖刻画。
英文摘要
An upward planar drawing of a directed acyclic graph is a planar drawing where every edge is pointed upward from its tail to head. Upward planar drawings are among the most natural drawing styles of directed graphs and have been researched in a variety of different settings, recently including that of drawing extension. In the drawing extension setting, one asks: given a graph $G$ and a (typically connected) subgraph $H$ of $G$ with a drawing $Γ(H)$, can we complete $Γ(H)$ to a drawing of $G$? Drawing extension problems have been studied for numerous drawing styles; the vast majority of these are NP-hard and a typical approach aimed at circumventing their intractability is to design parameterized algorithms where the parameter measures "how much" of $G$ is still missing from the pre-drawn graph $H$. Most algorithms obtained within this framework require only a small number of edges to be missing from $H$ in order to remain efficient. In this article, we present a fixed-parameter algorithm for extending upward planar drawings which overcomes this drawback by using the $\textit{vertex+edge deletion distance}$ as the parameter, thus achieving tractability even for instances with many missing edges. A key ingredient towards our result is a novel characterization of "canonical" sets of missing edges which cross a horizontal line segment in the drawing.
