Planar graph extends to a triangulated planar supergraph
ProvedWilliamsonShmoys.planar_supergraph_with_euler_edge_count__p2m_24bfc568_7b25_4409_b2ae_c59e72f324da_a1approximation-algorithmsplanar-graphstreewidth
Every finite drawable simple graph on at least three vertices can be extended, on the same vertex type, to a drawable graph with exactly 3|V|−6 edges. The extension is the triangulation step needed by the parent proof.
Preamble
import Mathlib import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
Formal statement
theorem WilliamsonShmoys.planar_supergraph_with_euler_edge_count__p2m_24bfc568_7b25_4409_b2ae_c59e72f324da_a1 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) (hplanar : WilliamsonShmoys.HasPlanarDrawingOn M) (hcard : 3 ≤ Fintype.card V) : ∃ N : SimpleGraph V, M ≤ N ∧ WilliamsonShmoys.HasPlanarDrawingOn N ∧ 3 * Fintype.card V = N.edgeSet.ncard + 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).