AI 中文总结
研究双连通直线平面绘图的部分绘图可扩展性问题（PDE），证明特定条件下PDE仍NP难，给出H为双连通、G有固定嵌入且含p条长度为2路径时的O(p^2 n)时间算法，还表明该问题在特定条件下关于G的顶点覆盖数是固定参数可处理的。
AI 中文摘要
部分绘图可扩展性问题（PDE）的输入为三元组⟨G, H, Γ_H⟩，其中G是平面图，H是G的子图，Γ_H是H的直线平面绘图，问题是Γ_H能否扩展为G的直线平面绘图。Patrignani证明PDE问题是NP难的。本文研究初始部分绘图Γ_H为双连通时的PDE问题。证明即使H是有界大小面的双连通图、G是次立方图且G中不在H中的部分由长度为2的路径组成，PDE仍为NP难。当G有固定嵌入且H连通（甚至双连通）时，PDE复杂度未知。作为解决此问题的一步，研究了H为双连通、G有固定嵌入且其余部分由p条长度为2的路径组成的PDE实例，给出了O(p^2 n)时间算法，还表明若H为双连通，该问题在固定和可变嵌入设置下关于G的顶点覆盖数是固定参数可处理的。
英文摘要
The Partial Drawing Extensibility problem, for short PDE, takes as input a triple $\langle G,H,Γ_H\rangle$, where $G$ is a planar graph, $H$ is a subgraph of $G$, and $Γ_H$ is a straight-line planar drawing of $H$, and asks whether $Γ_H$ can be extended to a straight-line planar drawing of $G$. Patrignani [Int. J. Found. Comput. Sci. (2006)] proved that the PDE problem is NP-hard, exploiting instances in which $H$ is highly disconnected. In this paper, we study the PDE problem under the requirement that the initial partial drawing $Γ_H$ is biconnected. We show that PDE remains NP-hard even for instances in which $H$ is a biconnected graph with faces of bounded size, $G$ is subcubic, and the part of $G$ that is not in $H$ consists of length-$2$ paths. The complexity of PDE remains however open when $H$ is connected (or even biconnected) if $G$ has a fixed embedding. In this setting both a polynomial-time algorithm or an NP-hardness proof seem to be elusive targets. As a step towards tackling this problem, we study instances of PDE in which $H$ is biconnected, $G$ has a fixed embedding, and the rest of the graph consists of $p$ length-2 paths, and present an $O(p^2 n)$-time algorithm, a result in sharp contrast with the NP-hardness of the variable embedding setting. Moreover, with an approach based on the Existential Theory of the Reals, we show that, if $H$ is biconnected, the problem is FPT parameterized by the vertex cover number of $G$, both in a fixed and in a variable embedding setting.
CommentsAppears in the Proceedings of the 34th International Symposium on Graph Drawing and Network Visualization
