Edge count of a maximal planar graph

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_61dc9438_8340_4e4c_b1c7_250f3c9d3d5d_a1

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

approximation-algorithmsplanar-graphstreewidth

A finite simple graph with at least three vertices that is maximal among planar supergraphs on the same vertex set has exactly 3|V| - 6 edges. The child removes the planar topology and face-incidence argument from the parent; the parent only needs the supplied Euler equation and linear arithmetic.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_61dc9438_8340_4e4c_b1c7_250f3c9d3d5d_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 <= Fintype.card V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hmax : forall L : SimpleGraph V, K <= L -> WilliamsonShmoys.HasPlanarDrawingOn L -> L <= K) :
    K.edgeSet.ncard + 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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