Edge count of an inclusion-maximal planar drawing

Proved

WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_dee0ca40_6c9c_434d_8ef7_a3ab7a09b0fc_a1

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

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal drawable simple graph on at least three vertices has exactly 3|V|−6 edges. The proof route is to show maximality makes every face triangular, then combine Euler’s formula with the face-edge incidence count.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_dee0ca40_6c9c_434d_8ef7_a3ab7a09b0fc_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hV : 3 <= Fintype.card V)
    (hM : HasPlanarDrawingOn M)
    (hmax : ∀ N : SimpleGraph V, M ≤ N → HasPlanarDrawingOn N → N = M) :
    (letI : DecidableRel M.Adj := Classical.decRel M.Adj
     3 * Fintype.card V = M.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).

View graph