The problem of extending partial geometric graph representations such as plane graphs has received considerable attention in recent years. In particular, given a graph $G$, a connected subgraph $H$ of $G$ and a drawing $\mathcal{H}$ of $H$, the extension problem asks whether $\mathcal{H}$ can be extended into a drawing of $G$ while maintaining some desired property of the drawing (e.g., planarity). In their breakthrough result, Angelini et al. [ACM TALG 2015] showed that the extension problem is polynomial-time solvable when the aim is to preserve planarity. Very recently we considered this problem for partial 1-planar drawings [ICALP 2020], which are drawings in the plane that allow each edge to have at most one crossing. The most important question identified and left open in that work is whether the problem can be solved in polynomial time when $H$ can be obtained from $G$ by deleting a bounded number of vertices and edges. In this work, we answer this question positively by providing a constructive polynomial-time decision algorithm.
翻译:近年来,扩大部分几何图形(如平面图)的问题引起了相当的注意。特别是,考虑到一张GG$,一个连接的Subalog $G$,一个连接的Subpology $G$,和一个绘图$gmathcal{H}$H美元,扩展问题问,美元是否可以在保持绘制的某些预期属性(例如,平面图)的同时扩大到G$的绘图。在其突破性结果中,Angelini et al.[ACM TALG 2015] 表明,扩展问题在旨在保持平面时是可以多时间隔的。最近,我们考虑了部分一平面图(CICP 2020)的问题,这是平面上绘制的,使每个边缘在大多数一个交叉处都能画出。在这项工作中发现和留下的最重要问题是,当通过删除一个约束的脊椎和边缘数来从$G时,能否在多边时间里解决问题。在这项工作中,我们通过提供建设性的多元时算法来正面解这个问题。