Triangulation extension of a finite planar drawing
ProvedWilliamsonShmoys.maximal_planar_drawing_triangulation_extension__p2m_0a0c7b94_7c6b_416a_9a27_7e1438879d42_a1approximation-algorithmsplanar-graphstreewidth
Every finite drawable simple graph on at least three vertices is contained in a drawable graph on the same vertices whose edges satisfy the planar triangulation count 3|V|−6.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_triangulation_extension__p2m_0a0c7b94_7c6b_416a_9a27_7e1438879d42_a1
{V : Type} [Fintype V] [LinearOrder V]
(R : SimpleGraph V) (hV : 3 <= Fintype.card V)
(hR : HasPlanarDrawingOn R) :
∃ T : SimpleGraph V, R ≤ T ∧ HasPlanarDrawingOn T ∧
(letI : DecidableRel T.Adj := Classical.decRel T.Adj
3 * Fintype.card V = T.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).