Finite planar graphs have a maximal planar supergraph

Proved

WilliamsonShmoys.planar_maximal_extension__p2m_2f35ea45_99df_4cf1_95b2_81f89b32ddea_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite graph with a drawing has a same-vertex planar supergraph that has no strictly larger planar supergraph on that vertex type. This is obtained by maximizing the number of edges among the finite collection of planar supergraphs containing the input graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_maximal_extension__p2m_2f35ea45_99df_4cf1_95b2_81f89b32ddea_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hH : HasPlanarDrawingOn H)
    (hn : 3 <= Fintype.card V) :
    ∃ K : SimpleGraph V, H ≤ K ∧ HasPlanarDrawingOn K ∧
      ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L = K := 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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