An inclusion-maximal drawable supergraph exists
ProvedWilliamsonShmoys.exists_inclusion_maximal_drawable_supergraph__p2m_78cb0e01_1105_4561_a4db_7810010c8df8_a1approximation-algorithmsplanar-graphstreewidth
Every finite drawable graph has a drawable supergraph on the same vertices that has no strict drawable supergraph.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_inclusion_maximal_drawable_supergraph__p2m_78cb0e01_1105_4561_a4db_7810010c8df8_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (dG : DecidableRel G.Adj)
(hG : HasPlanarDrawingOn G) :
∃ (Q : SimpleGraph V) (dQ : DecidableRel Q.Adj),
HasPlanarDrawingOn Q ∧ G ≤ Q ∧
(∀ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
HasPlanarDrawingOn K → Q ≤ K → K = Q) := 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).