A connected finite graph contains enough edges
ProvedWilliamsonShmoys.connected_edge_card_lower_bound__p2m_078d72d1_c4b8_4d42_8ead_9d2688891836_a1approximation-algorithmsplanar-graphstreewidth
Every finite connected simple graph has at least one fewer edge than vertices, as witnessed by a spanning tree.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_edge_card_lower_bound__p2m_078d72d1_c4b8_4d42_8ead_9d2688891836_a1 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) [DecidableRel G.Adj] (hconn : G.Connected) : Fintype.card V <= G.edgeFinset.card + 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).