Extend a planar drawing to a triangulation on the same vertices
ProvedWilliamsonShmoys.maximal_planar_triangulation_extension__p2m_4a0e4ba9_45c9_4065_a5b6_23412bcd7af0_a1approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on at least three vertices has a drawable supergraph on the same vertex set with exactly 3n−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_triangulation_extension__p2m_4a0e4ba9_45c9_4065_a5b6_23412bcd7af0_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hK : HasPlanarDrawingOn K) :
∃ L : SimpleGraph V, K ≤ L ∧ HasPlanarDrawingOn L ∧
(letI : DecidableRel L.Adj := Classical.decRel L.Adj;
3 * Fintype.card V = L.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).