Every planar graph has an edge-maximal planar completion

Proved

WilliamsonShmoys.exists_edge_maximal_planar_completion__p2m_92efd15a_1c37_4034_81a1_98dbd8d1208a_a1

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

Any graph with the specified planar drawing has an edge-inclusion-maximal planar supergraph on the same finite vertex type. Finiteness gives a maximal choice among the planar supergraphs containing the original graph; the drawing is preserved when edges are removed.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_edge_maximal_planar_completion__p2m_92efd15a_1c37_4034_81a1_98dbd8d1208a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V)
    (hN : WilliamsonShmoys.HasPlanarDrawingOn N) :
    ∃ K : SimpleGraph V,
      N ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧
        ∀ L : SimpleGraph V,
          K ≤ L → WilliamsonShmoys.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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