Finite maximal drawable extension

Proved

WilliamsonShmoys.exists_edge_maximal_planar_extension__p2m_7aa49e78_992e_47a4_a47f_24fc3e2cd1e1_a2

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

approximation-algorithmsplanar-graphstreewidth

Every connected drawable graph on a finite vertex type has a drawable supergraph that is maximal under edge inclusion among drawable graphs on the same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_edge_maximal_planar_extension__p2m_7aa49e78_992e_47a4_a47f_24fc3e2cd1e1_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hconn : K.Connected)
    (hK : HasPlanarDrawingOn K) :
    letI : ∀ G : SimpleGraph V, DecidableRel G.Adj :=
      fun G => Classical.decRel G.Adj
    ∃ G : SimpleGraph V,
      K.edgeFinset ⊆ G.edgeFinset ∧
      HasPlanarDrawingOn G ∧
      ∀ H : SimpleGraph V,
        G.edgeFinset ⊆ H.edgeFinset →
        HasPlanarDrawingOn H →
        H.edgeFinset ⊆ G.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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