Edge lower bound for an edge-saturated drawable graph

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WilliamsonShmoys.edge_saturated_planar_graph_edge_lower_bound__p2m_af9692fd_71ca_4e9b_bb88_c2808556729a_a3

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

approximation-algorithmsplanar-graphstreewidth

A finite simple graph on at least three vertices has at least 3|V|−6 edges if it has a crossing-free drawing and no missing edge can occur in any drawable supergraph containing it.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edge_saturated_planar_graph_edge_lower_bound__p2m_af9692fd_71ca_4e9b_bb88_c2808556729a_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hblocked : ∀ u v, u ≠ v → ¬ H.Adj u v →
      ∀ K : SimpleGraph V, H ≤ K → K.Adj u v →
        WilliamsonShmoys.HasPlanarDrawingOn K → False) :
    3 * Fintype.card V ≤ H.edgeSet.ncard + 6 := 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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