A connected graph extension adds an edge to a disconnected graph
ProvedWilliamsonShmoys.connected_extension_has_new_edge__p2m_47afe67c_60e1_47f9_a634_1f5964e2e86e_a3approximation-algorithmsplanar-graphstreewidth
If a graph is a subgraph of a connected graph but is disconnected, the larger graph has an edge whose endpoints are distinct and which was not an edge of the smaller graph.
Preamble
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_extension_has_new_edge__p2m_47afe67c_60e1_47f9_a634_1f5964e2e86e_a3
{V : Type} [Fintype V]
(K M : SimpleGraph V)
(hKM : K ≤ M)
(hK : ¬ K.Connected)
(hM : M.Connected) :
∃ u v : V, u ≠ v ∧ ¬ K.Adj u v ∧ M.Adj u 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).