Maximal planar graphs have the exact edge count
ProvedWilliamsonShmoys.maximal_planar_drawing_exact_edge_count__p2m_71c4bf82_58f5_4c33_a47c_7b17640d4f06_a2approximation-algorithmsplanar-graphstreewidth
A finite drawable simple graph on at least three vertices that is maximal under drawable extension has exactly 3|V| − 6 edges. The geometric work is to establish connectedness and triangular faces for a maximal planar drawing, then apply Euler’s formula and count face-edge incidences; the |V| = 3 case is included.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_exact_edge_count__p2m_71c4bf82_58f5_4c33_a47c_7b17640d4f06_a2
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (hV : 3 <= Fintype.card V)
(hN : HasPlanarDrawingOn N)
(hmax : ∀ K : SimpleGraph V, N <= K → HasPlanarDrawingOn K → K = N) :
(letI : DecidableRel N.Adj := Classical.decRel N.Adj;
N.edgeFinset.card + 6 = 3 * Fintype.card V) := 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).