Triangulation completion of a finite planar graph
OpenWilliamsonShmoys.planar_triangulation_extension__p2m_61f827a4_90a1_438d_8856_22a36ff94a2a_a2approximation-algorithmsplanar-graphstreewidth
Every drawable finite simple graph on at least three vertices has a drawable planar supergraph on the same vertex type with exactly three times the number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_triangulation_extension__p2m_61f827a4_90a1_438d_8856_22a36ff94a2a_a2
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hplanar : HasPlanarDrawingOn G) :
∃ L : SimpleGraph V, G.edgeSet ⊆ L.edgeSet ∧
HasPlanarDrawingOn L ∧
L.edgeSet.ncard + 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).