Planar triangulation extension on a fixed vertex set
ProvedWilliamsonShmoys.edge_maximal_planarDrawingOn_triangulation_extension__p2m_bfa78551_440c_46c3_a993_a460d980df6b_a1approximation-algorithmsplanar-graphstreewidth
Every finite graph with a crossing-free drawing and at least three vertices has a planar supergraph on the same vertex set with exactly 3|V|−6 edges. This is the triangulation-extension theorem for the target's arc-based definition of a planar drawing. The claim covers the equality boundary |V| = 3.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.edge_maximal_planarDrawingOn_triangulation_extension__p2m_bfa78551_440c_46c3_a993_a460d980df6b_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (dH : DecidableRel H.Adj)
(hH : HasPlanarDrawingOn H)
(hn : 3 ≤ Fintype.card V) :
∃ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
HasPlanarDrawingOn K ∧
(letI : DecidableRel H.Adj := dH;
letI : DecidableRel K.Adj := dK;
H.edgeFinset ⊆ K.edgeFinset ∧
K.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).