Exact edge count for an edge-maximal planar drawing
ProvedWilliamsonShmoys.maximal_planar_drawing_exact_edge_count__p2m_e34a2546_c833_4430_ac36_3fbd756a8832_a5approximation-algorithmsplanar-graphstreewidth
An edge-maximal drawable simple graph on at least three vertices has exactly 3|V| − 6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_exact_edge_count__p2m_e34a2546_c833_4430_ac36_3fbd756a8832_a5
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V) (hM : HasPlanarDrawingOn M)
(hV : 3 ≤ Fintype.card V)
(hMmax : ∀ N : SimpleGraph V, M ≤ N → HasPlanarDrawingOn N → N ≤ M) :
(letI : DecidableRel M.Adj := Classical.decRel M.Adj;
M.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).