Finite maximal drawable completion

Open

WilliamsonShmoys.exists_maximal_planar_completion__p2m_55edef6f_d8c2_4514_97bb_024e1dcb6408_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite drawable graph has a drawable supergraph on the same vertices that is maximal among drawable supergraphs of the original graph. The child removes the finite maximization and preservation-of-drawing construction from the parent. Prove it by maximizing the edge count among the finite set of drawable supergraphs; any proper supergraph has strictly more edges. The drawing predicate is inherited directly for the selected maximizer. This does not establish the edge count.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_completion__p2m_55edef6f_d8c2_4514_97bb_024e1dcb6408_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L) :
    ∃ M : SimpleGraph V, L ≤ M ∧ WilliamsonShmoys.HasPlanarDrawingOn M ∧
      (∀ N : SimpleGraph V, L ≤ N → WilliamsonShmoys.HasPlanarDrawingOn N → N ≤ M) := 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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