We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class $\mathcal H$ of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in $\mathcal H$. Our main result is that there exists a geometric graph with $n$ vertices and $O(n \log n)$ edges that is universal for $n$-vertex forests; this extends to the geometric setting a well-known graph-theoretic result by Chung and Graham, which states that there exists an $n$-vertex graph with $O(n \log n)$ edges that contains every $n$-vertex forest as a subgraph. Our $O(n \log n)$ bound on the number of edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that, for every positive integer $h$, every $n$-vertex convex geometric graph that is universal for $n$-vertex outerplanar graphs has a near-quadratic number of edges, namely $\Omega_h(n^{2-1/h})$; this almost matches the trivial $O(n^2)$ upper bound given by the $n$-vertex complete convex geometric graph. Finally, we prove that there exists an $n$-vertex convex geometric graph with $n$ vertices and $O(n \log n)$ edges that is universal for $n$-vertex caterpillars.

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通用几何图形

我们介绍并研究了构造具有很少顶点和边并且对平面图或平面图的某些子类具有普遍性的几何图的问题；如果几何图包含 $\mathcal H$ 中每个图的嵌入（即无交叉绘图），则该几何图对于类 $\mathcal H$ 的平面图来说是 \emph{universal}。我们的主要结果是存在一个具有 $n$ 顶点和 $O(n \log n)$ 边的几何图，对于 $n$-顶点森林是通用的；这扩展到几何设置，这是 Chung 和 Graham 著名的图论结果，它指出存在一个 $n$-顶点图，其中 $O(n\log n)$ 边包含每个 $n$-顶点森林作为子图。我们在边数上的 $O(n \log n)$ 限制无法改进，即使允许超过 $n$ 个顶点。我们也证明，对于每一个正整数$h$，每一个$n$-顶点外平面图通用的$n$-顶点凸几何图都有一个接近二次的边数，即$\Omega_h(n^{2-1/h })$; 这几乎与 $n$-顶点完全凸几何图给出的微不足道的 $O(n^2)$ 上限相匹配。最后，我们证明存在一个 $n$-顶点凸几何图，它具有 $n$ 个顶点和 $O(n\log n)$ 条边，对于 $n$-顶点毛虫是通用的。