Edge count of a maximal planar graph
ProvedWilliamsonShmoys.maximal_planar_edge_count__p2m_89a3e2c0_b2b3_4ebc_a282_9a392a489cfc_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices that has no strict drawable supergraph on the same vertices has exactly 3|V|−6 edges. A proof can derive connectedness and triangular faces from maximality, then apply Euler's formula and count face-edge incidences. This is the topological graph-theory core remaining in the reduction; it is not proved here.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_89a3e2c0_b2b3_4ebc_a282_9a392a489cfc_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hV : 3 <= Fintype.card V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hmax : ∀ L : SimpleGraph V, K <= L → WilliamsonShmoys.HasPlanarDrawingOn L → L <= K) :
letI : DecidableRel K.Adj := Classical.decRel K.Adj
3 * Fintype.card V = K.edgeFinset.card + 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).