Maximal planar drawing edge lower bound

Proved

WilliamsonShmoys.maximal_planar_drawing_edge_lower_bound__p2m_401105e3_1529_4fdd_958a_7e12d53ff038_a1

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A graph on at least three vertices that has a drawing and is maximal among drawable supergraphs on the same vertex type has at least 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_lower_bound__p2m_401105e3_1529_4fdd_958a_7e12d53ff038_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hM : HasPlanarDrawingOn M)
    (hmax : ∀ N : SimpleGraph V, HasPlanarDrawingOn N → M ≤ N → N = M) :
    3 * Fintype.card V - 6 ≤
      (letI : DecidableRel M.Adj := Classical.decRel M.Adj
       M.edgeFinset.card) := by sorry

Source

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).

View graph