Existence of an inclusion-maximal planar supergraph
ProvedWilliamsonShmoys.exists_inclusion_maximal_planar_supergraph__p2m_724aac7b_31c9_4291_9b6d_4edefcb29b4d_a1approximation-algorithmsplanar-graphstreewidth
Every graph with a drawing on a finite ordered vertex type has a planar supergraph on the same vertices that is maximal under inclusion among planar supergraphs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_inclusion_maximal_planar_supergraph__p2m_724aac7b_31c9_4291_9b6d_4edefcb29b4d_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) [DecidableRel G.Adj]
(hG : HasPlanarDrawingOn G) :
∃ (H : SimpleGraph V), ∃ (dH : DecidableRel H.Adj),
HasPlanarDrawingOn H ∧ G ≤ H ∧
∀ (K : SimpleGraph V) [DecidableRel K.Adj],
HasPlanarDrawingOn K → H ≤ K → K ≤ H := 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).