Complete planar graph completion for fewer than three vertices
ProvedWilliamsonShmoys_planar_small_card_completion__p2m_d31de1f7_ef50_4693_bbf0_93f59f6d9afd_a1approximation-algorithmsplanar-graphstreewidth
On a finite linearly ordered vertex type with fewer than three vertices, complete any drawable simple graph to the complete graph; it remains drawable and has the stated binomial edge count.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys_planar_small_card_completion__p2m_d31de1f7_ef50_4693_bbf0_93f59f6d9afd_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (dH : DecidableRel H.Adj)
(hH : WilliamsonShmoys.HasPlanarDrawingOn H)
(hsmall : Fintype.card V < 3) :
∃ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
WilliamsonShmoys.HasPlanarDrawingOn K ∧
H ≤ K ∧
K.edgeFinset.card = (Fintype.card V).choose 2 := 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).