Extend a drawable graph to a maximal planar edge count

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WilliamsonShmoys.planar_drawing_edge_count_maximal_extension__p2m_8348399b_8cb6_45dd_ab73_26e7cc43a44d_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on at least three vertices is a subgraph of a drawable simple graph on the same vertices having exactly 3|V|−6 edges. This isolates the topological extension and Euler-counting work; once proved, edge-set monotonicity yields the target.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_edge_count_maximal_extension__p2m_8348399b_8cb6_45dd_ab73_26e7cc43a44d_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hM : WilliamsonShmoys.HasPlanarDrawingOn M)
    (hCard : 3 ≤ Fintype.card V) :
    ∃ H : SimpleGraph V,
      M.edgeSet ⊆ H.edgeSet ∧
      H.edgeSet.ncard = 3 * Fintype.card V - 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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