Connect a finite planar drawing without losing edges

Proved

WilliamsonShmoys.planar_drawing_connected_extension__p2m_1450b57d_ff3f_4f0e_9779_6941f1e67e4b_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite ordered vertex type with at least three vertices, any graph with the specified crossing-free drawing has a connected planar supergraph on the same vertices whose edge count is at least the original graph’s.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_connected_extension__p2m_1450b57d_ff3f_4f0e_9779_6941f1e67e4b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn K) :
    letI : ∀ G : SimpleGraph V, DecidableRel G.Adj := fun G => Classical.decRel G.Adj
    ∃ H : SimpleGraph V, K ≤ H ∧ H.Connected ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
      K.edgeFinset.card ≤ H.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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