Universal geometric graphs

Abstract

We extend the notion of universal graphs to a geometric setting. A geometric graph is universal for a class of planar graphs if it contains an embedding, that is, a crossing-free drawing, of every graph in. Our main result is that there exists a geometric graph with vertices and edges that is universal for -vertex forests; this generalises a well-known result by Chung and Graham, which states that there exists an (abstract) graph with vertices and edges that contains every -vertex forest as a subgraph. The upper bound of edges cannot be improved, even if more than vertices are allowed. We also prove that every -vertex convex geometric graph that is universal for -vertex outerplanar graphs has a near-quadratic number of edges, namely, for every positive integer; this almost matches the trivial upper bound given by the -vertex complete convex geometric graph. Finally, we prove that there exists an -vertex convex geometric graph with vertices and edges that is universal for -vertex caterpillars.