Do two strongly connected nodes constitute a cycle in directed graph?

Question

I'm seeing seemingly contradictory information about this around. On the one hand, it seems that for undirected graphs, you need at least 3 nodes to make a cycle. On the other other, I've also seen people differentiate between trivial and non-trivial cycles.

Let's say in your directed graph you have two nodes A and B with an edge from A to B and an edge from B to A. Do you have a cycle (A, B)? And does it always count as a cycle, or would some people consider it a cycle and others not?

Also, let's say you traverse that directed graph and hit A before you hit B. Is the edge (B, A) a back edge?

Answer 1

Like most things, you should only care about definition if it makes sense and useful in your context.

And in the context of strongly connected component, a nice property we would like to say is that:

Node $a$ and $b$ belongs to the same cycle if and only if $a$ and $b$ are strongly connected.

This nice statement only holds if you count $f\rightarrow g\rightarrow f$ as a cycle. So, we would like to count this as a cycle when talking about strongly connected components.