Connected finite graph edge lower bound
ProvedWilliamsonShmoys.connected_graph_card_le_edge_ncard_add_one__p2m_c14927fd_3bb2_4aa2_8e62_690259e960a9_a1approximation-algorithmsplanar-graphstreewidth
Every finite connected simple graph has at least one fewer edge than vertices. A spanning-tree argument or induction on the vertex count proves this generic graph fact and supplies the numerical bound used by the parent.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_graph_card_le_edge_ncard_add_one__p2m_c14927fd_3bb2_4aa2_8e62_690259e960a9_a1
{V : Type} [Fintype V]
(K : SimpleGraph V) (hconn : K.Connected) :
Fintype.card V ≤ K.edgeSet.ncard + 1 := 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).