Edge bounds for a maximal planar drawing

Open

WilliamsonShmoys.maximal_planar_drawing_edge_bounds__p2m_5f4b31de_07d6_4550_bc2e_2bb318dd0027_a1

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

approximation-algorithmsplanar-graphstreewidth

A maximal simple graph with a crossing-free drawing and at least three vertices has enough edges for |V| ≤ |E| + 2 and no more than the planar bound |E| + 6 ≤ 3|V|.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_bounds__p2m_5f4b31de_07d6_4550_bc2e_2bb318dd0027_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hplanar : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hmax : ∀ J : SimpleGraph V, H.edgeSet ⊆ J.edgeSet →
      WilliamsonShmoys.HasPlanarDrawingOn J → J.edgeSet = H.edgeSet)
    (hcard : 3 ≤ Fintype.card V) :
    Fintype.card V ≤ H.edgeSet.ncard + 2 ∧
      H.edgeSet.ncard + 6 ≤ 3 * Fintype.card V := 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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