Planar extension with the maximal planar edge count

Proved

WilliamsonShmoys.planar_triangulation_supergraph_exact_edge_count__p2m_00360c86_73a0_483f_8c2b_2f99e65d88dd_a1

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

approximation-algorithmsplanar-graphstreewidth

Any drawable finite simple graph with at least three vertices has a drawable supergraph on the same vertex set whose edge count is exactly three times the vertex count minus six.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_triangulation_supergraph_exact_edge_count__p2m_00360c86_73a0_483f_8c2b_2f99e65d88dd_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hM : HasPlanarDrawingOn M) :
    ∃ N : SimpleGraph V,
      M ≤ N ∧ HasPlanarDrawingOn N ∧
        (letI : DecidableRel N.Adj := Classical.decRel N.Adj;
          N.edgeFinset.card + 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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