Connected planar extension preserves the vertex set

Disproved

WilliamsonShmoys.drawing_connected_extension__p2m_a60c0160_4029_4185_bfe8_ff4b14f19470_a2

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

approximation-algorithmsplanar-graphstreewidth

A graph with the given crossing-free drawing has a connected planar supergraph on the same finite vertex set. The original edge count is no larger than the extended graph’s edge count, for every decidable adjacency relation on the extension.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.drawing_connected_extension__p2m_a60c0160_4029_4185_bfe8_ff4b14f19470_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (dL : DecidableRel L.Adj)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L) :
    (letI : DecidableRel L.Adj := dL
     ∃ H : SimpleGraph V,
       (∀ u v, L.Adj u v → H.Adj u v) ∧
       (∀ (dH : DecidableRel H.Adj),
         (letI : DecidableRel H.Adj := dH
          L.edgeFinset.card ≤ H.edgeFinset.card)) ∧
       H.Connected ∧ WilliamsonShmoys.HasPlanarDrawingOn H) := 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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