Triangulation extension of a planar drawing
ProvedWilliamsonShmoys.planar_drawing_triangulation_extension__p2m_55bcde83_43a5_4c62_ba9b_1744a34b8f86_a1approximation-algorithmsplanar-graphstreewidth
A crossing-free drawing of a finite simple graph with at least three vertices can be completed, on the same vertex set, to a simple graph whose edge set contains the original edges and whose edge count is exactly 3|V|−6.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawing_triangulation_extension__p2m_55bcde83_43a5_4c62_ba9b_1744a34b8f86_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) [DecidableRel G.Adj]
(hG : WilliamsonShmoys.HasPlanarDrawingOn G)
(hcard : 3 ≤ Fintype.card V) :
∃ (H : SimpleGraph V),
(∀ [DecidableRel H.Adj],
G.edgeFinset ⊆ H.edgeFinset ∧
H.edgeFinset.card = 3 * Fintype.card V - 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).