From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices

Abstract

A set \(P = H \cup \{w\}\) of \(n+1\) 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 \(2^n\) distinct order types that correspond to wheel sets, the number of frequency vectors is only about \(2^{n/2}\). 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 \(\mathbb {R}^d\) for any fixed d. The result is an \(O(n^{d-1})\) 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 \(O(n^d\log n)\). Based on our result about simplicial depth, we can compute the number of facets of the convex hull of \(n=d+k\) points in general position in \(\mathbb {R}^d\) in time \(O(n^{\max \{\omega ,k-2\}})\) where \(\omega \approx 2.373\), even though the asymptotic number of facets may be as large as \(n^k\).

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Notes

  1. 1.

    Here, \(\varphi (k)\) denotes Euler’s totient function, which counts the integers coprime to k that are at most k.

  2. 2.

    Sequence A007147 on OEIS (Online Encyclopedia of Integer Sequences).

  3. 3.

    Sequence A000016 on OEIS.

  4. 4.

    Sequence A001006 on OEIS.

  5. 5.

    Sequence A000108 on OEIS, which also counts abstract binary trees.

  6. 6.

    Sequence A001764 on OEIS, which, incidentally, also counts abstract ternary trees.

  7. 7.

    Sequence A006013 on OEIS, which also counts pairs of abstract ternary trees, implying the used identity since the left hand side is just the convolution of the counting sequence of abstract ternary trees.

  8. 8.

    The number \(\Box \) is also the number of crossings of the complete geometric graph on \(\widetilde{P}\), a quantity that has obtained special attention in connection with the so-called rectilinear crossing number of \(K_n\) (i.e., the smallest number of crossings in a straight line drawing of the complete graph in the plane).

  9. 9.

    Following [39], we add the requirement that the origin is the centroid, in contrast to, e.g., [26, Chap. 5.6].

  10. 10.

    For \(n = a_1 + a_2 + \dots + a_m\), we have \(\sum _{\{i,j,k\} \in {\left( {\begin{array}{c}[m]\\ 3\end{array}}\right) }} a_i a_j a_k = \frac{1}{6} \sum _{i=1}^m a_i (n-a_i) (n - 2a_i)\).

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Acknowledgements

The first author acknowledges support by a Schrödinger fellowship of the Austrian Science Fund (FWF): J-3847-N35.

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Correspondence to Manuel Wettstein.

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Dedicated to the memory of Ricky Pollack.

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Pilz, A., Welzl, E. & Wettstein, M. From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices. Discrete Comput Geom 64, 1067–1097 (2020). https://doi.org/10.1007/s00454-019-00147-1

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Keywords

  • Geometric graphs
  • Order types
  • Frequency vectors
  • Embracing triangles
  • Simplicial depth
  • Polytopes

Mathematics Subject Classification

  • 05A19
  • 05C30
  • 52C99