Finite maximal planar supergraph extension

Proved

WilliamsonShmoys.planar_supergraph_maximal_extension__p2m_14b8c227_b724_4de5_85b9_cdf4d661eb0f_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on at least three vertices has a drawable supergraph on the same vertices that is maximal among drawable supergraphs. The finite set of possible added edges gives a concrete route: repeatedly extend the drawing whenever a further edge can be added without losing planarity, and stop when no such extension remains.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_supergraph_maximal_extension__p2m_14b8c227_b724_4de5_85b9_cdf4d661eb0f_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hN : HasPlanarDrawingOn N) :
    ∃ P : SimpleGraph V, N ≤ P ∧ HasPlanarDrawingOn P ∧
      ∀ Q : SimpleGraph V, P ≤ Q → HasPlanarDrawingOn Q → Q ≤ P := 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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