Planar extension with sharp edge count

Proved

WilliamsonShmoys.planar_supergraph_with_sharp_edge_count__p2m_804de7f6_f862_47b4_add3_5edea203f43e_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite planar graph on at least three vertices has a planar supergraph on the same vertex set with exactly 3|V|−6 edges. The parent uses maximality to show that this extension is equal to the original graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_supergraph_with_sharp_edge_count__p2m_804de7f6_f862_47b4_add3_5edea203f43e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 <= Fintype.card V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    ∃ L : SimpleGraph V,
      K <= L ∧ WilliamsonShmoys.HasPlanarDrawingOn L ∧
        L.edgeSet.ncard + 6 = 3 * Fintype.card V := 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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