Every finite planar drawing extends to a maximal planar graph with exactly 3n−6 edges

Proved

WilliamsonShmoys.planar_drawing_has_exact_edge_extension__p2m_2d3bff7d_8a6a_45d8_b748_fea14c5aa101_a1

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

approximation-algorithmsplanar-graphstreewidth

Any drawable graph on a finite linearly ordered vertex set of size at least three has a drawable supergraph on the same vertices with exactly 3n−6 edges. The parent’s maximality hypothesis forces this supergraph to equal the original graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_has_exact_edge_extension__p2m_2d3bff7d_8a6a_45d8_b748_fea14c5aa101_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hdrawing : WilliamsonShmoys.HasPlanarDrawingOn H) :
    letI : ∀ K : SimpleGraph V, DecidableRel K.Adj := fun K => Classical.decRel K.Adj
    ∃ K : SimpleGraph V, H ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧
      3 * Fintype.card V - 6 = K.edgeFinset.card := 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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