Lower edge bound for a maximal planar drawing
OpenWilliamsonShmoys.maximalPlanarDrawingOn_edge_count_lower__p2m_341f0f70_acb5_465d_8404_ec7db43234c5_a1approximation-algorithmsplanar-graphstreewidth
A maximal drawable graph on at least three finite ordered vertices has at least 3|V|−6 edges. Maximality is with respect to drawable supergraphs on the same vertex type.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximalPlanarDrawingOn_edge_count_lower__p2m_341f0f70_acb5_465d_8404_ec7db43234c5_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hCard : 3 <= Fintype.card V)
(hmax : forall L : SimpleGraph V, K <= L -> WilliamsonShmoys.HasPlanarDrawingOn L -> L <= K) :
3 * Fintype.card V - 6 <= K.edgeSet.ncard := 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).