Planar graph completion to a triangulation
ProvedWilliamsonShmoys.planar_graph_has_triangulated_supergraph__p2m_abe63458_bbbb_493a_8df0_3e0f2c6fa377_a1approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on at least three vertices has a drawable supergraph on the same vertex type with exactly 3|V|−6 edges. The extension is obtained by adding edges until the embedding is triangulated, then counting edges using Euler’s formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_graph_has_triangulated_supergraph__p2m_abe63458_bbbb_493a_8df0_3e0f2c6fa377_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hH : HasPlanarDrawingOn H) :
∃ K : SimpleGraph V, H ≤ K ∧ HasPlanarDrawingOn K ∧
(letI : DecidableRel K.Adj := Classical.decRel K.Adj
3 * Fintype.card V = K.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).