Planar completion to a triangulation edge count
OpenWilliamsonShmoys.maximal_planar_triangulation_extension__p2m_077d3f0d_ff79_48f3_bde0_954cbd7dcf68_a2approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on at least three vertices has a drawable supergraph on the same vertex set with exactly 3n − 6 edges. This supplies the triangulated completion used by the parent.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_triangulation_extension__p2m_077d3f0d_ff79_48f3_bde0_954cbd7dcf68_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hH : HasPlanarDrawingOn H)
(hcard : 3 <= Fintype.card V) :
letI : forall K : SimpleGraph V, DecidableRel K.Adj :=
fun K => Classical.decRel K.Adj
exists J : SimpleGraph V, And (HasPlanarDrawingOn J)
(And (H <= J) (J.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).