AI 中文总结
研究带列表约束的点集可嵌入性问题，针对点集处于凸位置及一般情况给出算法和复杂度结论，包括NP - 难性、固定参数可处理算法等，还研究了优化和扩展变体的复杂度。
AI 中文摘要
判定给定图是否能在平面直线绘制中，使每个顶点放置在给定有限点集的某些点上，这就是点集可嵌入性问题，是图绘制中的经典问题。本文研究更一般的嵌入性问题，即每个顶点\(v\)的放置受限在允许点的列表\(L(v)\)中。首先研究给定的点集处于凸位置的情况，证明即使给定图是匹配且双标记（即每个顶点至多有2个允许点），此情况也是NP - 难的。积极方面，给出了两种有效算法用于给定图\(G\)连通（不一定双标记）的情况：若\(G\)有需遵循的组合嵌入，可在多项式时间解决问题；否则关于最大顶点度可在固定参数可处理时间解决。这回答了Frati等人的一个开放问题。接着关注更一般情况，即给定的点集不一定处于凸位置，证明双标记路径的NP - 难性；还给出双标记图关于顶点覆盖数的固定参数可处理算法。通过建立三标记设置下顶点覆盖数为2时的准NP - 难性以及顶点覆盖数为1且任意\(L\)时的多项式时间可解性来补充后一结果。最后研究优化和扩展变体，前者证明是APX - 难的，后者在自然扩展参数下提供参数化复杂度二分法。
英文摘要
Deciding whether a given graph admits a planar straight-line drawing where each vertex is placed on some point from a given finite point set is known as Point Set Embeddability and is a classical problem in graph drawing. In this paper, we study the more general embeddability question where the placement of each vertex $v$ is restricted to a list $L(v)$ of admissible points. We first study the case where the given point set is in convex position. We show that this case is NP-hard even if the given graph is a matching and bi-labeled, i.e., each vertex has at most 2 admissible points. On the positive side, we present two efficient algorithms for the case where the given graph $G$ is connected (and not necessarily bi-labeled): if $G$ is equipped with a combinatorial embedding that needs to be respected, we can solve the problem in polynomial time; otherwise we can solve it in FPT-time with regard to the maximum vertex degree. In particular, this answers an open question by Frati, Glisse, Lenhart, Liotta, Mchedlidze, and Nishat [GD'13]. We then turn our attention to the more general case where the given point set is not necessarily in convex position. Here, we show NP-hardness for bi-labeled paths; notably these graphs have a unique combinatorial embedding and maximum degree two. We also present an FPT-algorithm with respect to the vertex cover number for the special case of bi-labeled graphs. We complement this latter result by establishing paraNP-hardness in the tri-labeled setting for vertex cover number 2 and polynomial-time solvability for vertex cover number 1 and arbitrary $L$. Finally, we study optimization and extension variants, where we want to maximize the number of edges or extend a partial drawing, respectively. For the former, we show APX-hardness and for the latter, we provide a parameterized complexity dichotomy under natural extension parameters.
CommentsAppears in the Proceedings of the 34th International Symposium on Graph Drawing and Network Visualization (GD 2026); 29 pages, 12 figures; abstract shortened to meet arXiv's requirements
