Planar completion with the maximal planar edge count
OpenWilliamsonShmoys.planar_completion_edge_count__p2m_d5765fc6_2061_49c6_883d_a2bbbc9ebfd2_a4approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices has a drawable supergraph on the same vertex set, with a finite edge set, whose edge count is exactly three times the number of vertices minus six.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_completion_edge_count__p2m_d5765fc6_2061_49c6_883d_a2bbbc9ebfd2_a4 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) (hcard : 3 ≤ Fintype.card V) (hplanar : HasPlanarDrawingOn G) : ∃ (K : SimpleGraph V) (hKfin : Fintype K.edgeSet), G ≤ K ∧ HasPlanarDrawingOn K ∧ (@SimpleGraph.edgeFinset V K hKfin).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).