Triangulated planar supergraph with maximal edge count
ProvedWilliamsonShmoys.planar_triangulation_supergraph_edge_count__p2m_c52ee0f6_cd77_40cf_a2f0_4d029ef0159f_a2approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices is a subgraph of a drawable graph on the same vertices whose edge count is exactly 3|V|−6.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open scoped BigOperators
Formal statement
theorem WilliamsonShmoys.planar_triangulation_supergraph_edge_count__p2m_c52ee0f6_cd77_40cf_a2f0_4d029ef0159f_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
∃ H : SimpleGraph V, K ≤ H ∧
WilliamsonShmoys.HasPlanarDrawingOn H ∧
H.edgeSet.ncard = 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).