Edge count of an edge-maximal planar graph
ProvedWilliamsonShmoys.maximal_planar_edge_count__p2m_95f0e96c_588c_4293_a881_4eae330a335f_a1approximation-algorithmsplanar-graphstreewidth
A finite simple graph on at least three vertices that has a planar drawing and is maximal among planar supergraphs on those vertices has exactly 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_95f0e96c_588c_4293_a881_4eae330a335f_a1
{V : Type} [Fintype V] [LinearOrder V]
(T : SimpleGraph V)
(hT : WilliamsonShmoys.HasPlanarDrawingOn T)
(hmax : ∀ U : SimpleGraph V,
T.edgeSet ⊆ U.edgeSet →
(∃ e, e ∈ U.edgeSet ∧ e ∉ T.edgeSet) →
¬ WilliamsonShmoys.HasPlanarDrawingOn U)
(hcard : 3 ≤ Fintype.card V) :
T.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).