Every drawable finite graph has a maximal drawable extension
ProvedWilliamsonShmoys.planar_graph_has_maximal_drawable_extension__p2m_df568397_14d2_4b43_b482_47a2ce432eea_a2approximation-algorithmsplanar-graphstreewidth
Every drawable finite simple graph is contained in a drawable graph on the same vertices that has no strictly larger drawable supergraph.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_graph_has_maximal_drawable_extension__p2m_df568397_14d2_4b43_b482_47a2ce432eea_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hH : HasPlanarDrawingOn H) :
∃ K : SimpleGraph V, H.edgeSet ⊆ K.edgeSet ∧ HasPlanarDrawingOn K ∧
∀ L : SimpleGraph V, K.edgeSet ⊆ L.edgeSet → HasPlanarDrawingOn L →
L.edgeSet ⊆ K.edgeSet := 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).