Maximal drawable graph reaches the triangulation edge count
OpenWilliamsonShmoys.maximal_planar_graph_edge_lower_bound__p2m_319863c5_d9cf_422f_8cc1_e579df5fd0de_a2approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal drawable simple graph on at least three vertices has at least three times its number of vertices minus six edges. The proof route is to use maximality to force connectedness and triangular faces in the given drawing, then count face-edge incidences and apply Euler’s formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_edge_lower_bound__p2m_319863c5_d9cf_422f_8cc1_e579df5fd0de_a2
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V)
(hCard : 3 ≤ Fintype.card V)
(hM : WilliamsonShmoys.HasPlanarDrawingOn M)
(hMax : ∀ L : SimpleGraph V,
M.edgeSet ⊆ L.edgeSet →
WilliamsonShmoys.HasPlanarDrawingOn L →
M.edgeSet = L.edgeSet) :
3 * Fintype.card V - 6 ≤ M.edgeSet.ncard := 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).