Universal Geometric Graphs
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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 universal for a class ℋ of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in 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 Ω_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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