Edge count of an inclusion-maximal drawable graph
ProvedWilliamsonShmoys.maximal_planar_graph_edge_count__p2m_c8fc71ba_c41f_4610_9114_f9df2f968aaf_a3approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal drawable simple graph on at least three vertices has exactly 3n − 6 edges. The new argument needed is the planar triangulation count: derive a cell decomposition from the drawing, show maximality forces triangular faces, then combine Euler’s formula with double counting of face-edge incidences. Formalizing that topological passage for the supplied continuous-arc drawing predicate is the remaining core obligation.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_c8fc71ba_c41f_4610_9114_f9df2f968aaf_a3
{V : Type} [Fintype V] [LinearOrder V]
(P : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hP : HasPlanarDrawingOn P)
(hmax : ∀ Q : SimpleGraph V,
(∀ u v, P.Adj u v → Q.Adj u v) →
HasPlanarDrawingOn Q →
(∀ u v, Q.Adj u v → P.Adj u v)) :
3 * Fintype.card V = P.edgeSet.ncard + 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).