Extend a planar drawing to the maximal planar edge count

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WilliamsonShmoys.planar_graph_extend_to_maximal_formula__p2m_f926186e_6933_42b8_b492_59096ed06c95_a1

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

approximation-algorithmsplanar-graphstreewidth

Every graph with a crossing-free drawing on a finite ordered vertex type of size at least three extends to a drawable supergraph on the same vertices whose edge count is exactly three times the number of vertices minus six.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_graph_extend_to_maximal_formula__p2m_f926186e_6933_42b8_b492_59096ed06c95_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hM : HasPlanarDrawingOn M)
    (hcard : LE.le 3 (Fintype.card V)) :
    ∃ N : SimpleGraph V, M ≤ N ∧ HasPlanarDrawingOn N ∧
      (letI : ∀ K : SimpleGraph V, DecidableRel K.Adj :=
        fun K => Classical.decRel K.Adj
       N.edgeFinset.card + 6 = 3 * Fintype.card V) := 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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