Edge-maximal drawable graph has the Euler edge count
ProvedWilliamsonShmoys.edge_maximal_drawable_has_euler_edge_count__p2m_f1c5cd14_5ded_4838_bd8a_1ed332d55054_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three finite linearly ordered vertices that is maximal among drawable supergraphs on those vertices has exactly 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.edge_maximal_drawable_has_euler_edge_count__p2m_f1c5cd14_5ded_4838_bd8a_1ed332d55054_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hH : HasPlanarDrawingOn H)
(hMax : ∀ (K : SimpleGraph V) [DecidableRel K.Adj],
H.edgeFinset ⊆ K.edgeFinset → HasPlanarDrawingOn K →
K.edgeFinset ⊆ H.edgeFinset)
(hV : 3 <= Fintype.card V) :
H.edgeFinset.card = 3 * Fintype.card V - 6 := 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).