标号点集上的相容路
Compatible Paths on Labelled Point Sets
摘要：
令$P$和$Q$是$\mathbb{R}^2$中具有相同基数的有限点集，每个点集标记为从$1$到$n$。分别跨越$P$和$Q$的两个不相交几何图$G_P$和$G_Q$称为相容的，如果对于$G_P$中的每个面$f$，在$G_Q$中存在一个对应的面，其边界上的顶点的顺时针排序与$f$中的相同。特别地，$G_P$和$G_Q$必须是同一个连通$n$ -顶点图的直线嵌入。判定两个标号点集是否容许相容的几何路是NP完全的。我们给出多项式时间算法，找到兼容的路径或报告，没有存在于三种情况下：$O(n)$时间点凸位置; $O(n^2)$时间为两个简单的多边形，其中的路径被限制为保持在封闭的多边形内;和$O(n^2 \log n)$时间点在一般位置，如果路径被限制为单调。
英文摘要：
Let $P$ and $Q$ be finite point sets of the same cardinality in $\mathbb{R}^2$, each labelled from $1$ to $n$. Two noncrossing geometric graphs $G_P$ and $G_Q$ spanning $P$ and $Q$, respectively, are called compatible if for every face $f$ in $G_P$, there exists a corresponding face in $G_Q$ with the same clockwise ordering of the vertices on its boundary as in $f$. In particular, $G_P$ and $G_Q$ must be straight-line embeddings of the same connected $n$-vertex graph. Deciding whether two labelled point sets admit compatible geometric paths is known to be NP-complete. We give polynomial-time algorithms to find compatible paths or report that none exist in three scenarios: $O(n)$ time for points in convex position; $O(n^2)$ time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and $O(n^2 \log n)$ time for points in general position if the paths are restricted to be monotone.
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