Triangulated drawable supergraph edge count

Proved

WilliamsonShmoys.drawable_graph_has_triangulated_supergraph_edge_count__p2m_33702fa3_4ca0_4ee2_9d3c_676a8b8ff565_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 is a subgraph of a drawable graph on the same vertices whose edge count is exactly 3|V|−6.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.drawable_graph_has_triangulated_supergraph_edge_count__p2m_33702fa3_4ca0_4ee2_9d3c_676a8b8ff565_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hn : 3 <= Fintype.card V) :
    ∃ K : SimpleGraph V, ∃ dK : DecidableRel K.Adj,
      WilliamsonShmoys.HasPlanarDrawingOn K ∧
      H ≤ K ∧
      (letI : DecidableRel K.Adj := dK;
       3 * Fintype.card V - 6 = K.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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