Existence of an edge maximal drawable supergraph

Proved

WilliamsonShmoys.edge_maximal_planar_supergraph__p2m_aea893ff_524d_432f_a94f_a28f6a185e5e_a2

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on the given finite vertex type is contained in a drawable supergraph with no proper drawable supergraph on those vertices. Prove this by maximizing the edge count over the finite collection of drawable supergraphs containing G.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.edge_maximal_planar_supergraph__p2m_aea893ff_524d_432f_a94f_a28f6a185e5e_a2 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) (hCard : 3 ≤ Fintype.card V) (hG : HasPlanarDrawingOn G) : ∃ K : SimpleGraph V, G ≤ 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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