Iterate planar cross-component edge extensions

Proved

WilliamsonShmoys.planarDrawing_connected_supergraph_from_edge_step__p2m_dddaee3a_0582_44d9_9566_a5dd1d115710_a1

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

approximation-algorithmsplanar-graphstreewidth

If each disconnected planar graph on the given finite vertex type admits a planar extension by one edge between distinct components, then iterating that step produces a connected planar supergraph of the original graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planarDrawing_connected_supergraph_from_edge_step__p2m_dddaee3a_0582_44d9_9566_a5dd1d115710_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hn : 3 ≤ Fintype.card V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hstep : ∀ G : SimpleGraph V,
      WilliamsonShmoys.HasPlanarDrawingOn G →
      ¬ G.Connected →
      ∃ u v : V, ¬ G.Reachable u v ∧
        ∃ H : SimpleGraph V,
          G ≤ H ∧
          H.edgeSet = G.edgeSet ∪ {Sym2.mk u v} ∧
          WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ H : SimpleGraph V,
      K ≤ H ∧ 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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