Edge count of an inclusion-maximal drawable graph

Proved

WilliamsonShmoys.edge_count_of_maximal_planar_graph__p2m_b4fc0ec6_9094_46c1_821a_02fa395e6854_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal drawable finite simple graph with at least three vertices has exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

Formal statement

theorem WilliamsonShmoys.edge_count_of_maximal_planar_graph__p2m_b4fc0ec6_9094_46c1_821a_02fa395e6854_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hN : WilliamsonShmoys.HasPlanarDrawingOn N)
    (hmax : ∀ K : SimpleGraph V, N ≤ K →
      WilliamsonShmoys.HasPlanarDrawingOn K → K = N) :
    N.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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