Completion of a drawable graph to a triangulation-sized supergraph
ProvedWilliamsonShmoys.planar_completion_with_exact_edge_count__p2m_126d21f0_5b15_4e81_98d4_afc1d569f5a6_a1approximation-algorithmsplanar-graphstreewidth
Every graph with the specified crossing-free drawing on at least three vertices has a drawable supergraph on the same vertex type whose edge count is exactly three times the number of vertices minus six.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_completion_with_exact_edge_count__p2m_126d21f0_5b15_4e81_98d4_afc1d569f5a6_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hV : 3 <= Fintype.card V)
(hG : HasPlanarDrawingOn G) :
∃ K : SimpleGraph V, G ≤ K ∧ HasPlanarDrawingOn K ∧
(letI : DecidableRel K.Adj := Classical.decRel K.Adj
K.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).