Edge count of a maximal planar drawing
ProvedWilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_8963bb62_e14c_4266_badc_3102e502f664_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices that has no proper drawable simple supergraph on the same vertices has exactly 3|V| - 6 edges. The independent proof route is to show maximality forces connectedness and triangular faces, then combine Euler's formula with the fact that each edge is incident to two face-sides. This isolates the topological edge-count argument from the parent. The lower boundary |V| = 3 is included and yields three edges; the hypothesis 3 ≤ |V| is needed for the maximal-planar edge-count result. This child contains the remaining core topological obligation.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_8963bb62_e14c_4266_badc_3102e502f664_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hH : HasPlanarDrawingOn H)
(hV : 3 <= Fintype.card V)
(hMax : forall J : SimpleGraph V, HasPlanarDrawingOn J ->
(forall u v, H.Adj u v -> J.Adj u v) ->
(forall u v, J.Adj u v -> H.Adj u v)) :
H.edgeFinset.card = 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).