Competitively Constructed Planar Graphs
Wesley Pegden, Eric Wang
TL;DR
This work studies two competitive frameworks for constructing planar graphs—the edge drawing and circle packing games—and develops sharp structural and asymptotic results about which planar graphs can be forced under various biases. It reveals a fundamental distinction: the circle packing game can realize arbitrarily large Apollonian networks and unbounded diameter, while the edge drawing game imposes diameter bounds and excludes certain minors (e.g., octahedron, pentagonal prism). The analysis combines constructive strategies (phase-based plans, active-face invariants) with classical circle-packing theory (Koebe–Andreev–Thurston) to derive both necessity (Apollonian subgraphs) and sufficiency (bias-enabled embeddings) results, and it culminates in several compelling questions about the limits and separations between the two games. The results contribute to understanding how local competitive moves can shape global planar structures and connect combinatorial game theory with geometric packing concepts.
Abstract
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.