Edge count of a maximal drawable graph

Proved

WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_bef9e736_66c4_4a44_9d5c_f3c0ae5dba68_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph that is inclusion-maximal among drawable graphs on the same vertex type, with at least three vertices, has exactly 3|V|−6 edges. The proof obligation is the topological core: establish connectedness and triangular faces, then combine Euler’s formula with face-edge incidence counting.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_bef9e736_66c4_4a44_9d5c_f3c0ae5dba68_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hM : HasPlanarDrawingOn M)
    (hV : 3 ≤ Fintype.card V)
    (hMax : ∀ K : SimpleGraph V, M ≤ K → HasPlanarDrawingOn K → K ≤ 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).

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