Competitively Constructed Planar Graphs

We introduce and study two Maker-Breaker-like games for constructing planar graphs: the edge drawing game, where two players take turns drawing non-intersecting edges between points in the plane, and the circle packing game, where the players take turns placing disjoint circles in the plane. Both games produce planar graphs: the edge drawing game results in a plane graph drawing, and the circle packing game yields a planar graph via the contact graph of the packing. For both games, we give necessary conditions under which a given planar graph can be constructed. We also show that the two games are indeed different by giving a class of graphs which can be constructed in one but not the other.