Edge count of a maximal planar graph
ProvedWilliamsonShmoys.edge_maximal_planar_edge_count__p2m_9aae8c1a_0088_4039_a6d9_b99fa0bd700a_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three finite ordered vertices that is maximal under inclusion among drawable graphs on the same vertices has exactly 3n−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.edge_maximal_planar_edge_count__p2m_9aae8c1a_0088_4039_a6d9_b99fa0bd700a_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V)
(hN : WilliamsonShmoys.HasPlanarDrawingOn N)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ K : SimpleGraph V,
N ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K = N) :
N.edgeSet.ncard = 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).