Maximal drawable supergraph on the same vertices

Proved

WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_c9a4a27a_018e_4f14_8723_166da166188f_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices has a drawable supergraph on the same vertices that is maximal among drawable supergraphs. Its edge count is at least the original graph’s edge count.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_c9a4a27a_018e_4f14_8723_166da166188f_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hH : HasPlanarDrawingOn H)
    (hcard : 3 ≤ Fintype.card V) :
    letI : ∀ K : SimpleGraph V, DecidableRel K.Adj :=
      fun K => Classical.decRel K.Adj
    ∃ M : SimpleGraph V,
      H ≤ M ∧
      H.edgeFinset.card ≤ M.edgeFinset.card ∧
      HasPlanarDrawingOn M ∧
      ∀ N : SimpleGraph V, M ≤ N → 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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