Edge count of a maximal simple planar graph
ProvedWilliamsonShmoys.maximal_planar_edge_count__p2m_63f4596f_16e9_46f5_8d7e_b4c7e853b2c9_a1__p2m_63f4596f_16e9_46f5_8d7e_b4c7e853b2c9_a2approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices that is maximal among drawable supergraphs on the same vertices has exactly 3n − 6 edges. The proof route is to show maximality forces a triangulation, then apply Euler’s formula and count edge-face incidences; the hypothesis n ≥ 3 is essential to the stated exact formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_63f4596f_16e9_46f5_8d7e_b4c7e853b2c9_a1__p2m_63f4596f_16e9_46f5_8d7e_b4c7e853b2c9_a2
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hG : HasPlanarDrawingOn G)
(hmax : ∀ K : SimpleGraph V, G ≤ K → HasPlanarDrawingOn K → K = G) :
G.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).