A drawable graph has a maximal drawable supergraph

Proved

WilliamsonShmoys.drawable_graph_has_euler_extension_exists_maximal_extension__p2m_76f2ce3c_d450_4358_bc4a_f7783646ef08_a2

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

approximation-algorithmsplanar-graphstreewidth

On a finite linearly ordered vertex type, any drawable graph has a drawable supergraph maximal among drawable supergraphs on the same vertices. The proof route is finite maximization of edge cardinality among drawable supergraphs. The assumption of at least three vertices is preserved to match the parent, though maximality itself does not require it.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.drawable_graph_has_euler_extension_exists_maximal_extension__p2m_76f2ce3c_d450_4358_bc4a_f7783646ef08_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hK : HasPlanarDrawingOn K) :
    ∃ L : SimpleGraph V, K ≤ L ∧ HasPlanarDrawingOn L ∧
      ∀ M : SimpleGraph V, L ≤ M → 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).

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