A disconnected drawn graph has a strictly larger planar supergraph

Disproved

WilliamsonShmoys.disconnected_planar_supergraph_strict_extension__p2m_8335b04f_4e67_4cd5_9960_e59cc2d55a11_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite linearly ordered vertex type, any disconnected graph with a crossing-free drawing has a planar supergraph on the same vertices with strictly more edges. The drawing of the supergraph may be chosen anew.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.disconnected_planar_supergraph_strict_extension__p2m_8335b04f_4e67_4cd5_9960_e59cc2d55a11_a1
    {V : Type} [Fintype V] [LinearOrder V] (N : SimpleGraph V)
    (hN : HasPlanarDrawingOn N) (hDisconnected : ¬ N.Connected) :
    ∃ K : SimpleGraph V, N ≤ K ∧ HasPlanarDrawingOn K ∧
      (letI : DecidableRel K.Adj := Classical.decRel K.Adj
       letI : DecidableRel N.Adj := Classical.decRel N.Adj
       N.edgeFinset.card < 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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