Extend a planar drawing to connected finite face data

Proved

WilliamsonShmoys.planarDrawingOn_to_planarMapDataExtension__p2m_9d34f080_ed68_445b_bb08_ad40d7fc216e_a1

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

approximation-algorithmsplanar-graphstreewidth

A crossing-free drawing of a finite graph with at least three vertices can be extended, by adding edges on the same vertex set, to a connected planar graph carrying the supplied finite face data. The output graph contains every original edge. The size hypothesis preserves the target’s boundary cases, including disconnected graphs and isolated vertices.

Preamble

import Definitions.Def_PlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planarDrawingOn_to_planarMapDataExtension__p2m_9d34f080_ed68_445b_bb08_ad40d7fc216e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (J : SimpleGraph V) [DecidableRel J.Adj]
    (hJ : WilliamsonShmoys.HasPlanarDrawingOn J)
    (hcard : 3 <= Fintype.card V) :
    ∃ (K : SimpleGraph V) (hK : DecidableRel K.Adj),
      (∀ ⦃u v : V⦄, J.Adj u v → K.Adj u v) ∧
      Nonempty (@PlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3 V (inferInstance) K hK) := 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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