Edge count of a maximal drawable graph
ProvedWilliamsonShmoys.edge_count_of_maximal_planar_drawing__p2m_7021208f_8613_4620_af29_8109890d733a_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal drawable simple graph on at least three vertices has exactly three times the number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
Formal statement
theorem WilliamsonShmoys.edge_count_of_maximal_planar_drawing__p2m_7021208f_8613_4620_af29_8109890d733a_a1
{V : Type} [Fintype V] [LinearOrder V]
(T : SimpleGraph V) (hT : WilliamsonShmoys.HasPlanarDrawingOn T)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ U : SimpleGraph V, T ≤ U → WilliamsonShmoys.HasPlanarDrawingOn U → U = T) :
letI : DecidableRel T.Adj := Classical.decRel T.Adj
T.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).