Triangulated planar extension on the same vertices
ProvedWilliamsonShmoys.exists_planar_triangulation_extension__p2m_3b1a98e5_f0d3_4027_b8f6_4e4c1f675a8e_a1approximation-algorithmsplanar-graphstreewidth
Every connected graph with a crossing-free drawing and at least three vertices has a planar supergraph on the same vertices whose edge count is three times the number of vertices minus six. The edge-count assertion is stated for any decidable adjacency instance on the extension.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_planar_triangulation_extension__p2m_3b1a98e5_f0d3_4027_b8f6_4e4c1f675a8e_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) [DecidableRel N.Adj]
(hN : WilliamsonShmoys.HasPlanarDrawingOn N)
(hconn : N.Connected)
(hcard : 3 ≤ Fintype.card V) :
∃ P : SimpleGraph V, N ≤ P ∧
WilliamsonShmoys.HasPlanarDrawingOn P ∧
∀ [DecidableRel P.Adj],
P.edgeFinset.card + 6 = 3 * Fintype.card V := by sorrySource
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).