Edge count of a maximal planar graph
OpenWilliamsonShmoys.edge_count_of_maximal_planar_graph__p2m_102f4bfa_075e_4b5b_84e9_4b6c34bfa40c_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three vertices whose edge count is maximal among drawable supergraphs on the same vertices has exactly 3|V|−6 edges. The proof route is to derive connectedness and triangular faces from maximality and apply Euler’s formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
Formal statement
theorem WilliamsonShmoys.edge_count_of_maximal_planar_graph__p2m_102f4bfa_075e_4b5b_84e9_4b6c34bfa40c_a1
{V : Type} [Fintype V] [LinearOrder V]
(P : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hP : WilliamsonShmoys.HasPlanarDrawingOn P)
(hmax : ∀ Q : SimpleGraph V,
(∀ u v, P.Adj u v → Q.Adj u v) →
WilliamsonShmoys.HasPlanarDrawingOn Q →
Q.edgeSet.ncard ≤ P.edgeSet.ncard) :
3 * Fintype.card V = P.edgeSet.ncard + 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).