Dense drawable supergraph extension

Proved

WilliamsonShmoys.planar_graph_has_dense_planar_supergraph__p2m_646b4c8c_0244_4cea_b3b0_947eb5ae9f81_a1

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

approximation-algorithmsplanar-graphstreewidth

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

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_graph_has_dense_planar_supergraph__p2m_646b4c8c_0244_4cea_b3b0_947eb5ae9f81_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V)
    (hN : WilliamsonShmoys.HasPlanarDrawingOn N)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ K : SimpleGraph V, N ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧
      3 * Fintype.card V - 6 ≤ K.edgeSet.ncard := 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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