Maximum edge-set cardinality implies maximality

Proved

WilliamsonShmoys.ncard_maximal_implies_maximal__p2m_f0ae2a89_338a_4b39_88cf_341f6f611f31_a2

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

approximation-algorithmsplanar-graphstreewidth

If every drawable supergraph of L has at most as many edges as L, then every drawable supergraph containing L equals L.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.ncard_maximal_implies_maximal__p2m_f0ae2a89_338a_4b39_88cf_341f6f611f31_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V)
    (hmax : ∀ M : SimpleGraph V, L ≤ M → WilliamsonShmoys.HasPlanarDrawingOn M →
      M.edgeSet.ncard ≤ L.edgeSet.ncard) :
    ∀ M : SimpleGraph V, L ≤ M → WilliamsonShmoys.HasPlanarDrawingOn M → M = L := 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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