A set $$P = H \cup \{w\}$$ P = H ∪ { w } of $$n+1$$ n + 1 points in general position in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on such a set P , it suffices to know the frequency vector of P . While there are roughly $$2^n$$ 2 n distinct order types that correspond to wheel sets, the number of frequency vectors is only about $$2^{n/2}$$ 2 n / 2 . We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, triangulations, and many more. Based on that, the corresponding numbers of graphs can be computed efficiently. In particular, we rediscover an already known formula for w -embracing triangles spanned by H . Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H , i.e., the number of w -embracing simplices. While our previous arguments in the plane do not generalize easily, we show how to use similar ideas in $$\mathbb {R}^d$$ R d for any fixed d . The result is an $$O(n^{d-1})$$ O ( n d - 1 ) time algorithm for computing the simplicial depth of a point w in a set H of n points, improving on the previously best bound of $$O(n^d\log n)$$ O ( n d log n ) . Based on our result about simplicial depth, we can compute the number of facets of the convex hull of $$n=d+k$$ n = d + k points in general position in $$\mathbb {R}^d$$ R d in time $$O(n^{\max \{\omega ,k-2\}})$$ O ( n max { ω , k - 2 } ) where $$\omega \approx 2.373$$ ω ≈ 2.373 , even though the asymptotic number of facets may be as large as $$n^k$$ n k .

中文翻译：

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从轮组上的无交叉图到包含很少顶点的单纯形和多面体

$$P = H \cup \{w\}$$ P = H ∪ { w } $$n+1$$ n + 1 个平面上一般位置的点被称为轮组，如果所有点都不是w是极端的。我们表明，为了计算这样一个集合 P 上的无交叉几何图，知道 P 的频率向量就足够了。虽然大约有 $$2^n$$ 2 n 种不同的订单类型对应于轮对，但频率向量的数量只有大约 $$2^{n/2}$$ 2 n / 2 。我们根据无交叉生成周期、匹配、三角剖分等的频率向量给出了简单的公式。在此基础上，可以有效地计算相应数量的图。特别是，我们重新发现了一个已知的公式，用于由 H 跨越的 w 包围三角形。同样在更高的维度上，轮组被证明是解决计算集合 H 中点 w 的单纯深度的问题的合适模型，即 w 包含单纯形的数量。虽然我们之前在平面上的论点不容易推广，但我们展示了如何在 $$\mathbb {R}^d$$ R d 中对任何固定的 d 使用类似的想法。结果是一个 $$O(n^{d-1})$$O ( nd - 1 ) 时间算法，用于计算 n 个点的集合 H 中点 w 的单纯深度，改进了之前的最佳边界$$O(n^d\log n)$$O ( nd log n ) 。基于我们关于单纯深度的结果，我们可以计算 $$n=d+k$$ n = d + k 点在 $$\mathbb {R}^d$$ 中的一般位置的凸包的面数R d 时间 $$O(n^{\max \{\omega ,k-2\}})$O ( n max { ω , k - 2 } ) 其中 $$\omega \approx 2.373$$ ω ≈ 2.373 ,