An inclusion-maximal drawable graph has maximum edge count
OpenWilliamsonShmoys.inclusion_maximal_planar_has_maximum_edges__p2m_78cb0e01_1105_4561_a4db_7810010c8df8_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal drawable simple graph on a finite vertex set has at least as many edges as any drawable simple graph on that set.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.inclusion_maximal_planar_has_maximum_edges__p2m_78cb0e01_1105_4561_a4db_7810010c8df8_a1
{V : Type} [Fintype V] [LinearOrder V]
(Q : SimpleGraph V) (dQ : DecidableRel Q.Adj)
(hQ : HasPlanarDrawingOn Q)
(hmaximal : ∀ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
HasPlanarDrawingOn K → Q ≤ K → K = Q)
(K : SimpleGraph V) (dK : DecidableRel K.Adj)
(hK : HasPlanarDrawingOn K) :
letI : DecidableRel Q.Adj := dQ
letI : DecidableRel K.Adj := dK
Q.edgeFinset.card ≤ K.edgeFinset.card := 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).