Existence of an edge-maximal planar supergraph

Proved

WilliamsonShmoys.exists_maximal_planar_supergraph_on_same_vertex_type__p2m_a4284538_7636_40d4_87e5_69297326a38c_a2

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

approximation-algorithmsplanar-graphstreewidth

Every finite graph with at least three vertices and a drawing of the specified kind is contained in a graph on the same vertex type that is maximal among its planar supergraphs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_supergraph_on_same_vertex_type__p2m_a4284538_7636_40d4_87e5_69297326a38c_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hdrawing : HasPlanarDrawingOn H) :
    ∃ K : SimpleGraph V, H ≤ K ∧ HasPlanarDrawingOn K ∧
      ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L ≤ K := 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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