Edge count of an inclusion-maximal planar drawing

Proved

WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_56b77430_5edc_41bf_8724_d4c1ab81c0d8_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph on at least three vertices that has no strictly larger drawable supergraph on the same vertex type has exactly 3|V| − 6 edges. Prove this by showing a maximal planar drawing is connected with triangular faces, then combining Euler’s formula with face-edge incidence counting.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_56b77430_5edc_41bf_8724_d4c1ab81c0d8_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V)
    (hcard : LE.le 3 (Fintype.card V))
    (hNdraw : HasPlanarDrawingOn N)
    (hmax : forall K : SimpleGraph V, LE.le N K -> HasPlanarDrawingOn K -> K = N) :
    3 * Fintype.card V =
      (letI : DecidableRel N.Adj := Classical.decRel N.Adj
       N.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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