What is the longest cycle of this digraph?

Question

Here's the diagram from my book:

I eliminated the vertices one at a time until I came up with my solution of $<b, c, g, f, d, b>$, but the book presents a cycle of equal length: $<d, b, c, g, f, d>$.

Now, observing those two cycles, they involve the same points, but they start and end at different vertices. Moreover, they're both the same length...

So is there in fact no cycle that is longer than any other, or are these two cycles the same (and therefore the longest), even though they start and end at different vertices?

Answer 1

I suppose it would depend on how you define 'same'. But for most purposes, I would say they are the same, after all, it is the same set of objects. It's like, are Alfred and Bob the same two people as Bob and Alfred? Of course, if for some reason the starting point did matter, than in that situation it would not be the same. I think the definition depends on the scenario, but in this one, they mean the same thing.

Answer 2

Ravi's comment alludes to the fact that both $\langle b,c,g,f,d,b \rangle$ and $\langle d,b,c,g,f,d \rangle$ are just different ways of writing the same cycle, namely the cycle obtained from the blue arrows in this modification of your diagram:

Importantly, both $\langle b,c,g,f,d,b \rangle$ and $\langle d,b,c,g,f,d \rangle$ use not only the same set of vertices, but also the same arrows, which is why we consider them to be the same cycle.