Complete the proof of Theorem 12.5. The following statements are equivalent for a loop-free undirected graph G

Question:

Complete the proof of Theorem 12.5.
The following statements are equivalent for a loop-free undirected graph G = (V, E).
(a) G is connected, and |V| = |E| + 1.
(b) G contains no cycles, and if a, b ∈ V with {a, b} ∉ E, then the graph obtained by adding edge {a, b} to G has precisely one cycle.