An inclusion-maximal planar graph has the triangulation edge count
OpenWilliamsonShmoys.maximal_planar_edge_count__p2m_8e688b5a_4cd6_4a71_af1c_8dd5b2ac94c3_a1approximation-algorithmsplanar-graphstreewidth
A finite inclusion-maximal planar simple graph with at least three vertices has exactly 3|V|−6 edges. The proof route is to use maximality to establish connectedness and triangular faces in a plane embedding, then apply Euler’s formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_8e688b5a_4cd6_4a71_af1c_8dd5b2ac94c3_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (dK : DecidableRel K.Adj)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hn : 3 ≤ Fintype.card V)
(hmax : ∀ (L : SimpleGraph V) (dL : DecidableRel L.Adj),
WilliamsonShmoys.HasPlanarDrawingOn L →
K.edgeFinset ⊆ L.edgeFinset → K.edgeFinset = L.edgeFinset) :
K.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).