In graph theory, a simple path is a path that contains no repeated vertices. But, in a directed graph, the directions of the arrows must be respected, right? That is A -> B <- C is not a path? However, I have a source which states that would also be a simple path, but, according to the same source, that would not be a directed path. Why would A -> B <- C be considered a simple path, or is the source wrong? So, what is a simple path in a directed graph?
[Math] a simple path in a directed graph
definitiongraph theory
Related Solutions
A loop is commonly defined as an edge (or directed edge in the case of a digraph) with both ends as the same vertex. (For example from $a$ to itself). Although loops are cycles, not all cycles are loops. In fact, none of the above digraphs have any loops.
Cycles are usually defined as closed walks which do not repeat edges or vertices except for the starting and ending vertex. This definition usually allows for cycles of length one (loops) and cycles of length two (parallel edges).
Note that cycles (and walks) do not make any reference to the orientation of the edges in question. Directed cycles (and directed walks) may only travel along the "forward" direction of the edges. In particular, that implies that $G_3$ pictured above has a third cycle, $(\color{blue}{(a,b)},(b,c),(c,a))$ where the $\color{blue}{(a,b)}$ refers instead to the edge pointing from $b$ to $a$.
Technically, all of the graphs above except for $G_2$ are directed multigraphs since in each you have parallel edges. Although in simple graphs (graphs with no loops or parallel edges) all cycles will have length at least $3$, a cycle in a multigraph can be of shorter length. Usually in multigraphs, we prefer to give edges specific labels so we may refer to them without ambiguity.
As for being strongly connected, yes all of them are and your definition is correct.
Your additional question, "what is the difference between a cycle and a connected component"
The above graph contains a cycle (though not a directed cycle) yet is not strongly connected.
One can prove that if a directed multigraph is strongly connected then it contains a cycle (take a directed walk from a vertex $v$ to $u$, then a directed walk from $u$ to $v$. Any closed walk contains a cycle).
One can also show that if you have a directed cycle, it will be a part of a strongly connected component (though it will not necessarily be the whole component, nor will the entire graph necessarily be strongly connected).
Yes, that is a simple directed graph (it has neither loops nor multiple arrows with the same source and target). However it is not a directed acyclic graph, because $u,v$ (or $1,3$ in your picture) form a directed cycle. See here for a description of many directed graph types.
Best Answer
There is not a quite universal consensus about the terminology here.
However, generally, most people would probably assume that when you have a directed graphs, the paths you're talking about will be directed path unless you're being quite explicit about ignoring the directionality.
I don't think just saying "simple" will be explicit enough to convey that. By that I would understand that the path cannot repeat vertices, but there's no reason such a path cannot be expected to respect the direction of edges -- so it makes plenty of sense to speak about either directed or undirected simple paths, and to implicitly assume the former when your edges happen to have direction.
If you want to speak about an undirected path in a directed graph, for the sake of communication please say "undirected path". If you need to decode something someone else says, you're at the mercy of their terminology. Ask for clarification if it is not clear what they mean.