An edge-maximal drawable graph has the planar maximum edge count

Proved

WilliamsonShmoys.planar_maximal_graph_edge_count__p2m_9bca3399_bc88_49f7_b83e_2aaaeb9cb6a3_a2

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

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph on at least three vertices that is maximal under drawable edge supergraphs has exactly 3|V| − 6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_maximal_graph_edge_count__p2m_9bca3399_bc88_49f7_b83e_2aaaeb9cb6a3_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG : HasPlanarDrawingOn G)
    (hV : 3 ≤ Fintype.card V)
    (hmax : ∀ (M : SimpleGraph V) [DecidableRel M.Adj],
      HasPlanarDrawingOn M →
      G.edgeFinset ⊆ M.edgeFinset →
      M.edgeFinset ⊆ G.edgeFinset) :
    G.edgeFinset.card = 3 * Fintype.card V - 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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