Edge count of an inclusion-maximal drawable graph

Proved

WilliamsonShmoys.inclusionMaximalPlanarSupergraph_has_maxEdges__p2m_541725ce_ba0e_4ec7_9bf8_a3e758b67715_a1

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

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal drawable simple graph with at least three vertices has exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.inclusionMaximalPlanarSupergraph_has_maxEdges__p2m_541725ce_ba0e_4ec7_9bf8_a3e758b67715_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : HasPlanarDrawingOn H)
    (hn : 3 ≤ Fintype.card V)
    (hmax : ∀ (J : SimpleGraph V) (_ : DecidableRel J.Adj),
      H ≤ J → HasPlanarDrawingOn J → J ≤ H) :
    3 * Fintype.card V - 6 = H.edgeFinset.card := 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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