Edge count of a maximal planar graph
OpenWilliamsonShmoys_edge_count_of_maximal_planar_supergraph__p2m_a21b9daf_96c0_42bf_b053_e1966758807a_a1approximation-algorithmsplanar-graphstreewidth
A connected, inclusion-maximal finite simple graph on at least three vertices that admits a drawing satisfying HasPlanarDrawingOn has exactly three times its number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys_edge_count_of_maximal_planar_supergraph__p2m_a21b9daf_96c0_42bf_b053_e1966758807a_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hcard : 3 ≤ Fintype.card V)
(hconn : H.Connected)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
(hmax : ∀ K : SimpleGraph V,
H ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K ≤ H) :
(letI : DecidableRel H.Adj := Classical.decRel H.Adj
H.edgeFinset.card + 6 = 3 * Fintype.card V) := 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).