Inclusion and edge cardinality determine equality
ProvedWilliamsonShmoys.eq_of_le_and_edge_card_bound__p2m_3721485b_3cdc_4a3e_9978_b518aa3e9dbd_a1approximation-algorithmsplanar-graphstreewidth
For graphs on a finite vertex type, if one graph contains another and its edge set has no greater cardinality, then the graphs are equal.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.eq_of_le_and_edge_card_bound__p2m_3721485b_3cdc_4a3e_9978_b518aa3e9dbd_a1
{V : Type} [Fintype V] [LinearOrder V]
(T U : SimpleGraph V) (hTU : T ≤ U)
(hcard : Nat.card U.edgeSet ≤ Nat.card T.edgeSet) : U = T := 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).