Smaller Induced-Universal Graph for Planar Graphs

A graph U is an induced-universal graph for a given family of graphs F if every graph of F is isomorphic to an induced-subgraph of U. A challenging question is to construct induceduniversal graphs of smallest size, i.e., with the minimum number of vertices. We construct an induced-universal graph of size n4/3+o(1) for the family of all n-vertex planar graphs. This is the first sub-quadratic bound for planar graphs. Our construction generalizes to the family of graphs of Euler genus at most g. The size of the universal graph is the same, up to a lower-order term depending on g.