Extend a connected planar graph to a maximal planar supergraph

Proved

WilliamsonShmoys_maximal_planar_extension_on_same_vertices__p2m_45533a44_b035_4e47_8a66_77352d0d9780_a1

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

approximation-algorithmsplanar-graphstreewidth

A connected graph with at least three vertices and a drawing satisfying the packaged planar-drawing predicate has a planar supergraph on the same vertices that is inclusion-maximal among simple planar supergraphs. Its edge count is at least that of the original graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys_maximal_planar_extension_on_same_vertices__p2m_45533a44_b035_4e47_8a66_77352d0d9780_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hconn : K.Connected)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    ∃ L : SimpleGraph V,
      K ≤ L ∧
      K.edgeSet.ncard ≤ L.edgeSet.ncard ∧
      WilliamsonShmoys.HasPlanarDrawingOn L ∧
      ∀ M : SimpleGraph V,
        L ≤ M → WilliamsonShmoys.HasPlanarDrawingOn M → M = L := 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).

View graph