Planar supergraph edge lower bound

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WilliamsonShmoys.planar_supergraph_edge_count__p2m_2caeaad5_0df9_4946_9307_3d000f81b17e_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph with at least three vertices has a drawable supergraph on the same vertex type whose edge count is at least three times the number of vertices minus six.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_supergraph_edge_count__p2m_2caeaad5_0df9_4946_9307_3d000f81b17e_a1 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) (hG : HasPlanarDrawingOn G) (hcard : 3 ≤ Fintype.card V) : ∃ S : SimpleGraph V, G ≤ S ∧ HasPlanarDrawingOn S ∧ ∀ [DecidableRel S.Adj], 3 * Fintype.card V - 6 ≤ S.edgeFinset.card := 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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