Existence of a maximal planar supergraph on the same vertices

Proved

WilliamsonShmoys_exists_maximal_planar_supergraph__p2m_a21b9daf_96c0_42bf_b053_e1966758807a_a1

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

approximation-algorithmsplanar-graphstreewidth

Every connected finite graph with a drawing satisfying HasPlanarDrawingOn has an inclusion-maximal planar supergraph on the same vertex type. Maximality is among all supergraphs on that type that also admit such a drawing; no vertex-count lower bound is needed for this extension step.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

Formal statement

theorem WilliamsonShmoys_exists_maximal_planar_supergraph__p2m_a21b9daf_96c0_42bf_b053_e1966758807a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hconn : G.Connected)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn G) :
    ∃ H : SimpleGraph V,
      G ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
        ∀ K : SimpleGraph V,
          H ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K ≤ 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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