Maximum-edge planar supergraph on a finite vertex type
ProvedWilliamsonShmoys.exists_planar_supergraph_max_edge_card__p2m_78898e31_8773_419c_99b2_d8837842d221_a2approximation-algorithmsplanar-graphstreewidth
Among all simple graphs on the same finite ordered vertex type that contain L and satisfy the given drawing predicate, one has at least as large an edge-set cardinality as every such graph. The drawing hypothesis on L makes this family nonempty. A proof can maximize edgeSet.ncard over the finite type of simple graphs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_planar_supergraph_max_edge_card__p2m_78898e31_8773_419c_99b2_d8837842d221_a2
{V : Type} [Fintype V] [LinearOrder V]
(L : SimpleGraph V)
(hL : HasPlanarDrawingOn L) :
∃ M : SimpleGraph V, L ≤ M ∧ HasPlanarDrawingOn M ∧
(∀ N : SimpleGraph V, L ≤ N → HasPlanarDrawingOn N →
N.edgeSet.ncard ≤ M.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).