Maximal drawable supergraph on the same vertices
ProvedWilliamsonShmoys.planar_drawing_has_maximal_planar_supergraph__p2m_98915fb2_fad3_4066_b195_28760c9e8707_a1approximation-algorithmsplanar-graphstreewidth
Every graph with the specified planar drawing on a finite ordered vertex type has a drawable supergraph on the same vertices that is maximal among drawable supergraphs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
Formal statement
theorem WilliamsonShmoys.planar_drawing_has_maximal_planar_supergraph__p2m_98915fb2_fad3_4066_b195_28760c9e8707_a1
{V : Type} [Fintype V] [LinearOrder V]
(Q : SimpleGraph V) (hQ : WilliamsonShmoys.HasPlanarDrawingOn Q) :
∃ R : SimpleGraph V, Q ≤ R ∧ WilliamsonShmoys.HasPlanarDrawingOn R ∧
∀ S : SimpleGraph V, R ≤ S → WilliamsonShmoys.HasPlanarDrawingOn S → S = R := 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).