Exact edge-count planar completion
ProvedWilliamsonShmoys.planar_drawing_exact_edge_count_completion__p2m_ecade84c_ffb2_4aac_a689_428692de6887_a1approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices has a drawable supergraph on the same vertex type whose edge count is exactly 3 times the number of vertices minus 6.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_exact_edge_count_completion__p2m_ecade84c_ffb2_4aac_a689_428692de6887_a1
{V : Type} [Fintype V] [LinearOrder V]
(Q : SimpleGraph V) (hQ : WilliamsonShmoys.HasPlanarDrawingOn Q)
(hcard : 3 ≤ Fintype.card V) :
∃ R : SimpleGraph V, Q ≤ R ∧ WilliamsonShmoys.HasPlanarDrawingOn R ∧
(letI : DecidableRel R.Adj := Classical.decRel R.Adj
3 * Fintype.card V = R.edgeFinset.card + 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).