Extend a planar drawing to a connected supergraph

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WilliamsonShmoys.planar_drawing_connected_supergraph__p2m_9739b138_86b0_425b_93f7_7e4f93a209fa_a4

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

approximation-algorithmsplanar-graphstreewidth

Any graph on the finite ordered vertex type that has a crossing-free drawing has a connected supergraph on that same vertex type with a crossing-free drawing. The result isolates the geometric extension and leaves edge-count monotonicity to a separate finite-graph lemma.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_connected_supergraph__p2m_9739b138_86b0_425b_93f7_7e4f93a209fa_a4
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L) :
    ∃ H : SimpleGraph V,
      (∀ u v, L.Adj u v → H.Adj u v) ∧
      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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