Extend a planar drawing to an edge-maximal planar graph

Proved

WilliamsonShmoys.planar_drawing_edge_maximal_extension__p2m_701795bb_5022_4e86_97bf_b07b90c6661c_a2

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

approximation-algorithmsplanar-graphstreewidth

Every finite graph with a planar drawing is contained in an inclusion-maximal planar graph on the same vertex type, and the extension has at least as many edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_edge_maximal_extension__p2m_701795bb_5022_4e86_97bf_b07b90c6661c_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V)
    (hP : WilliamsonShmoys.HasPlanarDrawingOn P)
    [DecidableRel P.Adj] :
    ∃ H : SimpleGraph V,
      P ≤ H ∧
      WilliamsonShmoys.HasPlanarDrawingOn H ∧
      (∀ K : SimpleGraph V,
        WilliamsonShmoys.HasPlanarDrawingOn K → H ≤ K → K ≤ H) ∧
      (∀ [DecidableRel H.Adj], P.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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