Planar drawing of a complete graph on fewer than three vertices
ProvedWilliamsonShmoys.small_card_complete_graph_planar_from_embedding__p2m_6ce4ca43_3f24_470a_9f5f_602e4f1a05ae_a1approximation-algorithmsplanar-graphstreewidth
An injective placement of fewer than three vertices in the plane induces a crossing-free drawing of the complete graph on those vertices, using straight-line arcs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.small_card_complete_graph_planar_from_embedding__p2m_6ce4ca43_3f24_470a_9f5f_602e4f1a05ae_a1
{V : Type} [Fintype V] [LinearOrder V]
(p : V → ℝ × ℝ) (hp : Function.Injective p)
(hsmall : Fintype.card V < 3) :
HasPlanarDrawingOn (⊤ : SimpleGraph V) := 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).