Edge count of a maximal drawable planar graph

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WilliamsonShmoys.maximal_planar_edge_count__p2m_aca1b5ba_75d7_404d_be10_c2c6ea433783_a1

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

approximation-algorithmsplanar-graphstreewidth

Every inclusion-maximal graph admitting the defined planar drawing, on at least three vertices, has exactly 3n − 6 edges. This removes the maximality-to-triangulation and planar edge-count argument from the parent. A proof route is to show maximality forces triangular faces in a planar embedding and then combine Euler’s formula with the fact that each edge contributes two face sides. The remaining core obligation is formalizing that planar graph argument under the given drawing definition.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_aca1b5ba_75d7_404d_be10_c2c6ea433783_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hM : HasPlanarDrawingOn M)
    (hcard : LE.le 3 (Fintype.card V))
    (hMax : forall N : SimpleGraph V, M <= N -> HasPlanarDrawingOn N -> N = M) :
    letI : forall K : SimpleGraph V, DecidableRel K.Adj :=
      fun K => Classical.decRel K.Adj
    M.edgeFinset.card + 6 = 3 * Fintype.card V := 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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