Maximal planar extension on a fixed vertex set

Proved

WilliamsonShmoys.planar_maximal_extension__p2m_1bf322db_0dbd_4aed_abd6_bf402bf7bb2f_a2

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

approximation-algorithmsplanar-graphstreewidth

Every graph with the given planar drawing is contained in an edge-maximal planar graph on the same finite vertex type. The maximality condition says that any planar supergraph on that type has the same edge set.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_maximal_extension__p2m_1bf322db_0dbd_4aed_abd6_bf402bf7bb2f_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hH : HasPlanarDrawingOn H) :
    ∃ K : SimpleGraph V,
      H.edgeSet ⊆ K.edgeSet ∧
      HasPlanarDrawingOn K ∧
      ∀ L : SimpleGraph V,
        K.edgeSet ⊆ L.edgeSet →
        HasPlanarDrawingOn L →
        L.edgeSet = K.edgeSet := 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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