Same-vertex-set triangulation completion
ProvedWilliamsonShmoys.planar_triangulation_completion__p2m_61543ccf_c10c_45ac_85dc_db241ebaf64e_a1approximation-algorithmsplanar-graphstreewidth
Every drawable finite simple graph on at least three vertices has a drawable supergraph on the same vertex set with exactly 3|V|−6 edges. The extension allows disconnected input graphs and isolated vertices.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_triangulation_completion__p2m_61543ccf_c10c_45ac_85dc_db241ebaf64e_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hcard : 3 <= Fintype.card V)
(hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
∃ K : SimpleGraph V, G <= K ∧
WilliamsonShmoys.HasPlanarDrawingOn K ∧
K.edgeSet.ncard = 3 * Fintype.card V - 6 := 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).