Planar supergraph with triangulated edge count
ProvedWilliamsonShmoys.exists_planar_supergraph_with_triangulated_edge_count__p2m_b6ab3cff_6690_4a88_8301_64964a0bb841_a1approximation-algorithmsplanar-graphstreewidth
Every finite drawable graph on at least three vertices has a drawable supergraph on the same vertex type with exactly three times the number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.exists_planar_supergraph_with_triangulated_edge_count__p2m_b6ab3cff_6690_4a88_8301_64964a0bb841_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hK : HasPlanarDrawingOn K)
(hV : 3 ≤ Fintype.card V) :
∃ L : SimpleGraph V, K ≤ L ∧ HasPlanarDrawingOn L ∧
(letI : DecidableRel L.Adj := Classical.decRel L.Adj
letI : Fintype ↑L.edgeSet := Fintype.ofFinite _
3 * Fintype.card V = L.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).