A drawable graph has a maximal drawable supergraph
ProvedWilliamsonShmoys.planar_drawing_maximal_supergraph_exists__p2m_71c4bf82_58f5_4c33_a47c_7b17640d4f06_a2approximation-algorithmsplanar-graphstreewidth
Every finite drawable graph is contained in a drawable supergraph maximal under inclusion. The argument maximizes edge count among drawable supergraphs of M; a strict extension would increase that count.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_maximal_supergraph_exists__p2m_71c4bf82_58f5_4c33_a47c_7b17640d4f06_a2
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V) (hM : HasPlanarDrawingOn M) :
Exists (fun N : SimpleGraph V =>
M <= N ∧ HasPlanarDrawingOn N ∧
∀ K : SimpleGraph V, N <= K → HasPlanarDrawingOn K → K = N) := by
sorrySource
David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).