Maximal planar supergraph completion on a fixed vertex set

Proved

WilliamsonShmoys.planar_supergraph_maximal_completion__p2m_76432751_4a60_438b_af25_c322d55d602e_a2

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

approximation-algorithmsplanar-graphstreewidth

Every finite planar simple graph with at least three vertices has a planar supergraph on the same vertex set that is inclusion-maximal among planar supergraphs on that set. Finiteness allows choosing a planar supergraph with a largest edge set.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_supergraph_maximal_completion__p2m_76432751_4a60_438b_af25_c322d55d602e_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (hV : 3 ≤ Fintype.card V) :
    ∃ H : SimpleGraph V, ∃ (dH : DecidableRel H.Adj),
      (letI : DecidableRel H.Adj := dH
       G.edgeFinset ⊆ H.edgeFinset ∧
       WilliamsonShmoys.HasPlanarDrawingOn H ∧
       ∀ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
         letI : DecidableRel K.Adj := dK
         WilliamsonShmoys.HasPlanarDrawingOn K →
         H.edgeFinset ⊆ K.edgeFinset → K.edgeFinset ⊆ H.edgeFinset) := 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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