Simple planar graph edge upper bound
ProvedWilliamsonShmoys.planar_edge_upper_bound__p2m_08c24848_031b_43ea_aa26_092fb49494c8_a2approximation-algorithmsplanar-graphstreewidth
A finite graph with at least three vertices and a drawing satisfying HasPlanarDrawingOn has at most 3|V| - 6 edges. This supplies the upper bound independently of maximality. A proof route is to obtain the planar Euler edge bound componentwise from the drawing, treating one- and two-vertex components directly.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_edge_upper_bound__p2m_08c24848_031b_43ea_aa26_092fb49494c8_a2
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V)
(hcard : 3 <= Fintype.card V)
(hM : HasPlanarDrawingOn M) :
(letI : (G : SimpleGraph V) -> DecidableRel G.Adj :=
fun G => Classical.decRel G.Adj
M.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).