Edge count of a maximal planar graph
ProvedWilliamsonShmoys.maximal_planar_graph_edge_count__p2m_60f46665_7c74_42d7_b1ac_33e75a012818_a1approximation-algorithmsplanar-graphstreewidth
An edge maximal drawable finite simple graph with at least three vertices has exactly 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_60f46665_7c74_42d7_b1ac_33e75a012818_a1
{V : Type} [Fintype V] [LinearOrder V]
(L : SimpleGraph V)
(hL : WilliamsonShmoys.HasPlanarDrawingOn L)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ M : SimpleGraph V, L.edgeSet ⊆ M.edgeSet →
WilliamsonShmoys.HasPlanarDrawingOn M → M.edgeSet = L.edgeSet) :
L.edgeSet.ncard = 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).