Edge count of a maximal drawable graph

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_2d6ff097_7c5b_40ac_9dcb_907c25524093_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable graph with at least three vertices that has no proper drawable supergraph on the same vertices has exactly 3|V|−6 edges. A proof route is to show maximality forces connectedness and triangular faces in a planar embedding, then combine the face-degree equation 3F=2E with Euler’s formula V−E+F=2. The boundary hypothesis is retained because the edge formula need not hold for smaller vertex sets.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_2d6ff097_7c5b_40ac_9dcb_907c25524093_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (Q : SimpleGraph V)
    (hcard : 3 <= Fintype.card V)
    (hQ : WilliamsonShmoys.HasPlanarDrawingOn Q)
    (hmax : ∀ R : SimpleGraph V,
      (∀ {u v : V}, Q.Adj u v → R.Adj u v) →
      WilliamsonShmoys.HasPlanarDrawingOn R →
      R = Q) :
    letI : DecidableRel Q.Adj := Classical.decRel Q.Adj
    3 * Fintype.card V = Q.edgeFinset.card + 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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