Connected planar graphs extend to a triangulation
ProvedWilliamsonShmoys.edgeMaximalPlanar_triangulatedExtension__p2m_07c9f31b_494b_4180_969f_c152b9627c5e_a1approximation-algorithmsplanar-graphstreewidth
Every connected drawable simple graph on a finite linearly ordered vertex type with at least three vertices has a drawable supergraph on the same vertices with exactly 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.edgeMaximalPlanar_triangulatedExtension__p2m_07c9f31b_494b_4180_969f_c152b9627c5e_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) [DecidableRel K.Adj]
(hK : HasPlanarDrawingOn K)
(hconn : K.Connected)
(hn : 3 ≤ Fintype.card V) :
∃ (H : SimpleGraph V) (_ : DecidableRel H.Adj),
K ≤ H ∧ HasPlanarDrawingOn H ∧
3 * Fintype.card V - 6 = H.edgeFinset.card := 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).