Connected simple graph edge bound below three vertices
ProvedWilliamsonShmoys.connected_graph_edge_bound_small__p2m_2921e68e_aa42_4dea_8127_751b3e371ec8_a1approximation-algorithmsplanar-graphstreewidth
A finite connected simple graph with fewer than three vertices satisfies the edge-count inequality by the generic simple-graph edge bound; connectivity excludes the empty vertex type.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_graph_edge_bound_small__p2m_2921e68e_aa42_4dea_8127_751b3e371ec8_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (dG : DecidableRel G.Adj)
(hconn : G.Connected)
(hcard : Fintype.card V < 3) :
(letI : DecidableRel G.Adj := dG; G.edgeFinset.card) + 3 ≤ 3 * Fintype.card V := 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).