Edge count of an edge-maximal drawable graph
OpenWilliamsonShmoys.maximal_planar_drawing_exact_edge_count__p2m_db3907fa_cf67_46ea_9385_43b8ffa0c261_a1approximation-algorithmsplanar-graphstreewidth
An edge-inclusion-maximal drawable simple graph on at least three vertices has exactly 3|V|−6 edges. The proof route is to show maximality forces connectedness and triangular faces, then combine Euler's formula with counting edge-face incidences. It contains the remaining topological core; the statement also covers |V|=3, where maximality forces the triangle.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_exact_edge_count__p2m_db3907fa_cf67_46ea_9385_43b8ffa0c261_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hH : WilliamsonShmoys.HasPlanarDrawingOn H)
(hCard : 3 ≤ Fintype.card V)
(hMax : ∀ N : SimpleGraph V,
H.edgeSet ⊆ N.edgeSet →
WilliamsonShmoys.HasPlanarDrawingOn N →
N.edgeSet ⊆ H.edgeSet) :
H.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).