Maximal drawable supergraph on the same vertices

Proved

WilliamsonShmoys.maximal_planar_completion_for_edge_bound__p2m_55c0bceb_ffb6_44c6_a362_90e494710496_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph on the fixed vertex type is contained in an inclusion-maximal drawable supergraph, with its edge count no larger than the supergraph’s.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_completion_for_edge_bound__p2m_55c0bceb_ffb6_44c6_a362_90e494710496_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ M : SimpleGraph V,
      H ≤ M ∧
      WilliamsonShmoys.HasPlanarDrawingOn M ∧
      (∀ N : SimpleGraph V, M ≤ N → WilliamsonShmoys.HasPlanarDrawingOn N → N ≤ M) ∧
      (letI : DecidableRel H.Adj := Classical.decRel H.Adj
       letI : DecidableRel M.Adj := Classical.decRel M.Adj
       H.edgeFinset.card ≤ M.edgeFinset.card) := 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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