A drawable extension of maximum edge count exists

Proved

WilliamsonShmoys.exists_card_maximal_planar_extension__p2m_5d6e3508_c11e_4337_a63b_6a8f12f05f58_a2

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

approximation-algorithmsplanar-graphstreewidth

Among the drawable supergraphs of N, one has edge-set cardinality at least that of every other drawable supergraph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_card_maximal_planar_extension__p2m_5d6e3508_c11e_4337_a63b_6a8f12f05f58_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hN : HasPlanarDrawingOn N) :
    ∃ K : SimpleGraph V, N ≤ K ∧ HasPlanarDrawingOn K ∧
      ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L →
        L.edgeSet.ncard ≤ K.edgeSet.ncard := 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).

View graph