Finite maximal drawable extension

Proved

WilliamsonShmoys.connected_planar_maximal_extension__p2m_f281671c_d623_4140_b8c6_78197a35da85_a1

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

approximation-algorithmsplanar-graphstreewidth

Every connected graph with the stated finite drawing and at least three vertices is a subgraph of an inclusion-maximal drawable simple graph on the same vertices, with its edge count no larger.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.connected_planar_maximal_extension__p2m_f281671c_d623_4140_b8c6_78197a35da85_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hG : HasPlanarDrawingOn G) (hconn : G.Connected)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ (H : SimpleGraph V) (dH : DecidableRel H.Adj),
      HasPlanarDrawingOn H ∧ G ≤ H ∧
      (letI : DecidableRel G.Adj := dG
       letI : DecidableRel H.Adj := dH
       G.edgeFinset.card ≤ H.edgeFinset.card) ∧
      (∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K = H) := 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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