Strict planar augmentation below the triangulation bound

Proved

WilliamsonShmoys.planar_drawing_strict_augmentation__p2m_3ca4615a_eebc_4af0_940e_7aeb5834c3ad_a1

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

approximation-algorithmsplanar-graphstreewidth

If a finite simple graph on at least three vertices has a drawing satisfying HasPlanarDrawingOn and fewer than 3|V|−6 edges, it has a strictly larger planar simple supergraph on the same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_strict_augmentation__p2m_3ca4615a_eebc_4af0_940e_7aeb5834c3ad_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hM : HasPlanarDrawingOn M)
    (hE : (letI : DecidableRel M.Adj := Classical.decRel M.Adj
           M.edgeFinset.card) < 3 * Fintype.card V - 6) :
    ∃ M' : SimpleGraph V, HasPlanarDrawingOn M' ∧ M ≤ M' ∧ M' ≠ M := 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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