Edge count of an inclusion-maximal drawable graph

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WilliamsonShmoys.maximal_drawable_edge_count__p2m_3a1649bb_b908_4400_8c80_435b14739a2c_a1

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

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal drawable simple graph on at least three vertices has exactly three times the number of vertices minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_drawable_edge_count__p2m_3a1649bb_b908_4400_8c80_435b14739a2c_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V)
    (hCard : 3 ≤ Fintype.card V)
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M)
    (hMax : ∀ L : SimpleGraph V,
      M.edgeSet ⊆ L.edgeSet →
      WilliamsonShmoys.HasPlanarDrawingOn L →
      M.edgeSet = L.edgeSet) :
    M.edgeSet.ncard = 3 * Fintype.card V - 6 := 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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