Extend a planar drawing to a triangulation
OpenWilliamsonShmoys.planar_drawing_extends_to_triangulation__p2m_e13857bb_dfd7_4e7a_a1e7_eab5feffe183_a3approximation-algorithmsplanar-graphstreewidth
A crossing-free drawing of a simple graph on at least three vertices extends on the same vertex set to a crossing-free supergraph with exactly 3|V|−6 edges. The child carries a decidable adjacency instance for the supergraph so its edge finset is well-formed.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_extends_to_triangulation__p2m_e13857bb_dfd7_4e7a_a1e7_eab5feffe183_a3 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) [DecidableRel L.Adj] (hL : HasPlanarDrawingOn L) (hcard : 3 ≤ Fintype.card V) : ∃ H : SimpleGraph V, ∃ (dH : DecidableRel H.Adj), (letI : DecidableRel H.Adj := dH; L.edgeFinset ⊆ H.edgeFinset ∧ HasPlanarDrawingOn H ∧ H.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).