Dense drawable supergraph augmentation

Proved

WilliamsonShmoys.planar_supergraph_edge_bound__p2m_09ee00d6_d6e0_4e98_aab0_c4cdc0b16fb7_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices has a drawable supergraph on the same vertex type with at least 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_supergraph_edge_bound__p2m_09ee00d6_d6e0_4e98_aab0_c4cdc0b16fb7_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hdraw : HasPlanarDrawingOn L) :
    letI : ∀ G : SimpleGraph V, DecidableRel G.Adj := fun G => Classical.decRel G.Adj
    ∃ M : SimpleGraph V,
      L.edgeFinset ⊆ M.edgeFinset ∧
      HasPlanarDrawingOn M ∧
      3 * Fintype.card V - 6 ≤ M.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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