Completion of a planar drawing to the maximal planar edge count
OpenWilliamsonShmoys.planar_drawing_completion_edge_count__p2m_899b0d31_5122_4616_b662_d79a2f207d1e_a1approximation-algorithmsplanar-graphstreewidth
Every drawable finite simple graph on at least three vertices can be extended, on the same vertex set, to a drawable graph with exactly 3|V| − 6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_completion_edge_count__p2m_899b0d31_5122_4616_b662_d79a2f207d1e_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hG : HasPlanarDrawingOn G)
(hcard : 3 <= Fintype.card V) :
∃ H : SimpleGraph V, G ≤ H ∧ HasPlanarDrawingOn H ∧
(letI : DecidableRel H.Adj := Classical.decRel H.Adj;
3 * Fintype.card V = H.edgeFinset.card + 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).