Existence of an inclusion-maximal planar supergraph on the same vertices

Proved

WilliamsonShmoys.exists_inclusion_maximal_planar_supergraph__p2m_bf9cfcbc_5b59_49b9_abb2_d36aa9cb22f2_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finitely drawn planar graph has a supergraph on the same vertices that is inclusion-maximal among graphs on those vertices admitting such a drawing.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_inclusion_maximal_planar_supergraph__p2m_bf9cfcbc_5b59_49b9_abb2_d36aa9cb22f2_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hM : HasPlanarDrawingOn M) :
    ∃ H : SimpleGraph V, M ≤ H ∧ HasPlanarDrawingOn H ∧
      ∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K ≤ H := 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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