Finite termination of strict planar extensions

Proved

WilliamsonShmoys.planar_drawing_connected_augmentation_of_step__p2m_a61b4d52_11af_49e1_8b44_bf89679ad338_a2

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

approximation-algorithmsplanar-graphstreewidth

If every disconnected planar supergraph of a finite graph has a strictly larger planar supergraph on the same vertices, then a connected planar supergraph containing the original graph exists.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_connected_augmentation_of_step__p2m_a61b4d52_11af_49e1_8b44_bf89679ad338_a2 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) (hL : HasPlanarDrawingOn L) (hstep : ∀ G : SimpleGraph V, L ≤ G → HasPlanarDrawingOn G → ¬ G.Connected → ∃ H : SimpleGraph V, G < H ∧ HasPlanarDrawingOn H) : ∃ H : SimpleGraph V, L ≤ H ∧ H.Connected ∧ 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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