Existence of a maximal planar supergraph

Proved

WilliamsonShmoys.planar_supergraph_maximal_extension_exists__p2m_2a77e1c0_132e_4864_a8ba_ceda5e75c647_a2

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

approximation-algorithmsplanar-graphstreewidth

Every finite connected graph with a crossing-free drawing has an inclusion-maximal planar supergraph on the same vertex type. The extension remains connected and has at least as many edges as the original graph.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_supergraph_maximal_extension_exists__p2m_2a77e1c0_132e_4864_a8ba_ceda5e75c647_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hplanar : HasPlanarDrawingOn G)
    (hconn : G.Connected) :
    Exists fun H : SimpleGraph V =>
      G <= H /\
      HasPlanarDrawingOn H /\
      G.edgeSet.ncard <= H.edgeSet.ncard /\
      H.Connected /\
      (forall K : SimpleGraph V, H <= K -> 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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