Edge count of a maximal planar supergraph
ProvedWilliamsonShmoys.maximal_planar_supergraph_edge_count__p2m_14b8c227_b724_4de5_85b9_cdf4d661eb0f_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three vertices that is maximal under drawable edge additions has exactly 3|V|−6 edges. A proof can establish that maximality forces triangular faces in the drawing, then combine Euler’s formula with face-edge incidence counting, including the three-vertex boundary case.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_supergraph_edge_count__p2m_14b8c227_b724_4de5_85b9_cdf4d661eb0f_a1
{V : Type} [Fintype V] [LinearOrder V]
(P : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hP : HasPlanarDrawingOn P)
(hmax : ∀ Q : SimpleGraph V, P ≤ Q → HasPlanarDrawingOn Q → Q ≤ P) :
(letI : DecidableRel P.Adj := Classical.decRel P.Adj;
3 * Fintype.card V = P.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).