
graph question
I didn't understand the meaning of a sentence in my textbook:
"In the same way (as any walk contains a path), one could argue that any closed walk contains a cycle, but there is one exception to be made: it is possible to walk from x_{1} to x_{n} and then walk back by retracing your steps. the graph of such a walk will not contain a cyle, unless the x_{1}x_{n} walk contained one. But any circuit must contain a cycle."
Question #1:
How is it not possible to walk back and forth from x_{1} and x_{n}?
I mean, if I imagine a graph with vertices and edges, if you can go from x_{1} to x_{n}, it seems pretty obvious to me that you can go from x_{n} to x_{1}.
Question #2:
Don't "closed walk" and "circuit" both mean "a walk with no repeated edges where the start and finish are the same"?
Then the text doesn't make sense because at first it's saying that not all closed walks contains a cycle, but in the end it's saying the opposite.