Same-vertex triangulation extension of a planar drawing
ProvedWilliamsonShmoys.planar_triangulation_extension__p2m_1cbebd7a_cbfe_40cb_8e61_899ac9ec9081_a3approximation-algorithmsplanar-graphstreewidth
Every finite simple graph with a crossing-free drawing and at least three vertices is contained on the same vertex set in a drawable graph with exactly 3|V|−6 edges. The child explicitly carries the decidable-adjacency instance needed to form the edge finset of the extended graph. The parent applies maximality to show the extension equals the original graph, yielding the target count.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_triangulation_extension__p2m_1cbebd7a_cbfe_40cb_8e61_899ac9ec9081_a3
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) [DecidableRel N.Adj]
(hN : HasPlanarDrawingOn N)
(hcard : 3 ≤ Fintype.card V) :
∃ K : SimpleGraph V, N ≤ K ∧ HasPlanarDrawingOn K ∧
∃ (hKd : DecidableRel K.Adj),
(letI : DecidableRel K.Adj := hKd;
K.edgeFinset.card + 6 = 3 * Fintype.card 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).