Triangulated completion of a planar drawing

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WilliamsonShmoys.maximal_planar_completion__p2m_f98f2276_465d_4fad_a092_9abf857bca84_a2

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

approximation-algorithmsplanar-graphstreewidth

Every graph with a crossing-free drawing on a finite linearly ordered vertex type of size at least three is contained in a graph on the same vertices that has a crossing-free drawing and exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_completion__p2m_f98f2276_465d_4fad_a092_9abf857bca84_a2 {V : Type} [Fintype V] [LinearOrder V] (H : SimpleGraph V) [DecidableRel H.Adj] (hH : HasPlanarDrawingOn H) (hcard : 3 <= Fintype.card V) : exists (K : SimpleGraph V) (hKdec : DecidableRel K.Adj) (hKfin : Fintype K.edgeSet), (forall u v, H.Adj u v -> K.Adj u v) /\ HasPlanarDrawingOn K /\ 3 * Fintype.card V = (@SimpleGraph.edgeFinset V K hKfin).card + 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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