  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.
