Augmenting a sparse planar graph

Proved

WilliamsonShmoys.planar_edge_augmentation__p2m_c0d4ecaf_266f_445a_8f51_dc327adf9453_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

If a drawable simple graph on at least three vertices has fewer than 3|V|−6 edges, it has a strictly larger drawable supergraph on the same vertex set.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open scoped BigOperators

Formal statement

theorem WilliamsonShmoys.planar_edge_augmentation__p2m_c0d4ecaf_266f_445a_8f51_dc327adf9453_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hsparse : K.edgeSet.ncard < 3 * Fintype.card V - 6) :
    ∃ L : SimpleGraph V,
      K ≤ L ∧ L ≠ K ∧ WilliamsonShmoys.HasPlanarDrawingOn L := 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).

View graph