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.

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在多项式时间内扩展几乎完整的 1 平面图

近年来，平面图等部分几何图形表示的扩展问题受到了相当多的关注。特别地，给定图$G$、$G$的连通子图$H$和$H$的图$\mathcal{H}$，扩展问题询问$\mathcal{H}$是否可以扩展到 $G$ 的绘图中，同时保持绘图的某些所需属性（例如，平面性）。在他们的突破性结果中，Angelini 等人。[ACM TALG 2015] 表明，当目标是保持平面性时，扩展问题是多项式时间可解的。最近，我们在部分 1 平面图 [ICALP 2020] 中考虑了这个问题，这些图是平面中的图，允许每条边最多有一个交叉点。在这项工作中确定并悬而未决的最重要的问题是，当可以通过删除有限数量的顶点和边从 $G$ 获得 $H$ 时，是否可以在多项式时间内解决问题。在这项工作中，我们通过提供一个建设性的多项式时间决策算法来积极地回答这个问题。