Maximal drawable supergraph on a fixed finite vertex set
ProvedWilliamsonShmoys.exists_maximal_planar_drawing_supergraph__p2m_7021208f_8613_4620_af29_8109890d733a_a1approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on a finite ordered vertex type is contained in an inclusion-maximal drawable graph on the same vertex set.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
Formal statement
theorem WilliamsonShmoys.exists_maximal_planar_drawing_supergraph__p2m_7021208f_8613_4620_af29_8109890d733a_a1
{V : Type} [Fintype V] [LinearOrder V]
(S : SimpleGraph V) (hS : WilliamsonShmoys.HasPlanarDrawingOn S) :
∃ T : SimpleGraph V,
S ≤ T ∧ WilliamsonShmoys.HasPlanarDrawingOn T ∧
∀ U : SimpleGraph V, T ≤ U → WilliamsonShmoys.HasPlanarDrawingOn U → U = T := 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).