Triangulated supergraph data from a planar drawing

Proved

WilliamsonShmoys.planar_drawing_has_triangulated_supergraph__p2m_c3950eb8_41cb_4e65_a38c_065736da1c7d_a5

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

approximation-algorithmsplanar-graphstreewidth

Extend the drawing to a maximal planar supergraph on the same vertex set and provide its face count, Euler relation, triangular-face incidence equation, and edge-count monotonicity. The parent uses this certificate to reduce the target to arithmetic. Prove it by planar augmentation and face counting; the topology of augmentation is the remaining core obligation. The hypothesis of at least three vertices is retained for this construction and the Euler/triangulation argument.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_has_triangulated_supergraph__p2m_c3950eb8_41cb_4e65_a38c_065736da1c7d_a5
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hcard : 3 <= Fintype.card V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ (K : SimpleGraph V) (hKdec : DecidableRel K.Adj) (f : Nat),
      (∀ ⦃u v : V⦄, H.Adj u v → K.Adj u v) ∧
      H.edgeFinset.card ≤ K.edgeFinset.card ∧
      K.edgeFinset.card + 2 = Fintype.card V + f ∧
      3 * f = 2 * 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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