I was hoping someone could help me understand this graph theorem better.
Theorem: Adding an edge from any graph G either joins two components of G or adds a cycle to G, but not both.
Especially this tidbit:
Proof: Let G be an arbitrary graph, let e = {u, v} be any edge that is not in G, and let G" = (V, E ∪ {e}).
• If u and v are in different components of G, those two components are joined in G" . If any cycle in G" contains edge e, then it contains another path from u to v, all of whose edges are in G, which is impossible. Thus, no cycle in G" contains edge e. It follows that every cycle in G" is also a cycle in G.
u and v are different vertices, and G'' is the graph of G but with one extra edge right? It says I can't create a cycle, but isn't this graph picture a counter example? The circles are vertexes. I'm guessing I'm misunderstanding the proof. Thanks for the help.