Edge count of a maximal planar graph
OpenWilliamsonShmoys.maximal_planar_edge_count__p2m_842d51ec_371b_408a_a7fe_9c9e23312f53_a1approximation-algorithmsplanar-graphstreewidth
A graph on at least three vertices that has the stated planar drawing and is maximal among planar supergraphs has exactly three times its number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_842d51ec_371b_408a_a7fe_9c9e23312f53_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hplanar : HasPlanarDrawingOn K)
(hmax : ∀ L : SimpleGraph V,
K.edgeSet ⊆ L.edgeSet →
HasPlanarDrawingOn L →
L.edgeSet = K.edgeSet) :
K.edgeSet.ncard + 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).