Extend a planar graph below the maximal edge count
OpenWilliamsonShmoys.planar_strict_extension_below_max__p2m_26401141_9acd_45a7_9799_c63f2391b494_a2__p2m_26401141_9acd_45a7_9799_c63f2391b494_a3approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices with fewer than 3n−6 edges has a drawable strict edge supergraph on the same vertex set.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_strict_extension_below_max__p2m_26401141_9acd_45a7_9799_c63f2391b494_a2__p2m_26401141_9acd_45a7_9799_c63f2391b494_a3
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hG : HasPlanarDrawingOn G)
(hcard : 3 ≤ Fintype.card V) :
letI : ∀ H : SimpleGraph V, DecidableRel H.Adj :=
fun H => Classical.decRel H.Adj
G.edgeFinset.card < 3 * Fintype.card V - 6 →
∃ H : SimpleGraph V,
G.edgeFinset ⊆ H.edgeFinset ∧
HasPlanarDrawingOn H ∧
¬ H.edgeFinset ⊆ G.edgeFinset := 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).