Edge lower bound for a maximal drawable graph

Proved

WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_dcda2b2e_bb79_4b65_906a_4b43dfb1e747_a1

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

approximation-algorithmsplanar-graphstreewidth

A connected graph on at least three vertices that is inclusion-maximal among drawable supergraphs on the same vertex type has at least three times the number of vertices minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_dcda2b2e_bb79_4b65_906a_4b43dfb1e747_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hN : HasPlanarDrawingOn N)
    (hmax : ∀ P : SimpleGraph V, N ≤ P → HasPlanarDrawingOn P → P = N)
    (hconn : N.Connected) :
    (letI : DecidableRel N.Adj := Classical.decRel N.Adj
     3 * Fintype.card V - 6 ≤ N.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).

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