Every planar drawing extends to an edge-maximal planar graph

Proved

WilliamsonShmoys.planar_maximal_extension__p2m_711cef84_2190_482b_bc2d_a11a812cbece_a1

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

approximation-algorithmsplanar-graphstreewidth

On a fixed finite vertex type, any graph with a planar drawing has a planar supergraph that is maximal under edge inclusion, and the supergraph has at least as many edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_maximal_extension__p2m_711cef84_2190_482b_bc2d_a11a812cbece_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hL : HasPlanarDrawingOn L) :
    ∃ H : SimpleGraph V,
      HasPlanarDrawingOn H ∧
      (letI : DecidableRel L.Adj := Classical.decRel L.Adj
       letI : DecidableRel H.Adj := Classical.decRel H.Adj
       L.edgeFinset.card ≤ H.edgeFinset.card) ∧
      (∀ K : SimpleGraph V, HasPlanarDrawingOn K →
        (∀ u v, H.Adj u v → K.Adj u v) →
        ∀ u v, K.Adj u v → H.Adj u v) := 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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