Strict planar edge extension below the maximal count

Proved

WilliamsonShmoys.WilliamsonShmoys_planar_graph_edge_extension__p2m_126bf562_f451_4e0b_b87e_ef5c21746e7b_a6

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

approximation-algorithmsplanar-graphstreewidth

If a drawable finite simple graph on at least three vertices has fewer than 3|V|−6 edges, it has a drawable strict supergraph on the same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.WilliamsonShmoys_planar_graph_edge_extension__p2m_126bf562_f451_4e0b_b87e_ef5c21746e7b_a6
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hP : HasPlanarDrawingOn P)
    (hlt : (letI : DecidableRel P.Adj := Classical.decRel P.Adj;
      P.edgeFinset.card + 6 < 3 * Fintype.card V)) :
    ∃ Q : SimpleGraph V, P ≤ Q ∧ ¬ Q ≤ P ∧ HasPlanarDrawingOn Q := 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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