Maximal drawable graph has the planar edge count
ProvedWilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_af412bdb_08d3_443e_9526_1f4e854292c2_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal drawable simple graph on at least three finite vertices has exactly three times the number of vertices minus six edges. This removes the planar edge-count step from the parent; the proof route is to derive connectedness and triangular faces from maximality, then combine Euler’s formula with the count of face-edge incidences for the given drawing definition.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_af412bdb_08d3_443e_9526_1f4e854292c2_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hM : HasPlanarDrawingOn M)
(hmax : ∀ N : SimpleGraph V, M ≤ N → HasPlanarDrawingOn N → N ≤ M) :
(letI : DecidableRel M.Adj := Classical.decRel M.Adj
M.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).