Planar graphs admit a triangulating supergraph
ProvedWilliamsonShmoys.planar_triangulation_completion__p2m_14be45b2_4af8_4477_b67a_6d2fa4a08c68_a2approximation-algorithmsplanar-graphstreewidth
Every drawable finite simple graph on at least three vertices has a drawable supergraph on the same vertices with exactly 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_triangulation_completion__p2m_14be45b2_4af8_4477_b67a_6d2fa4a08c68_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hcard : 3 ≤ Fintype.card V) :
∃ L : SimpleGraph V, K.edgeSet ⊆ L.edgeSet ∧
WilliamsonShmoys.HasPlanarDrawingOn L ∧
L.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).