Planar graphs extend to a triangulation
ProvedWilliamsonShmoys.planar_completion_with_maximal_edge_count__p2m_2f6efb11_cffe_4335_a64a_833cc169dca5_a2approximation-algorithmsplanar-graphstreewidth
Every drawable finite simple graph on at least three vertices is contained in a drawable graph on the same vertices with exactly 3|V| − 6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_completion_with_maximal_edge_count__p2m_2f6efb11_cffe_4335_a64a_833cc169dca5_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) [DecidableRel K.Adj]
(hK : HasPlanarDrawingOn K)
(hV : 3 ≤ Fintype.card V) :
∃ (L : SimpleGraph V), HasPlanarDrawingOn L ∧
∃ (dL : DecidableRel L.Adj),
(letI : DecidableRel L.Adj := dL;
K.edgeFinset ⊆ L.edgeFinset ∧
L.edgeFinset.card = 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).