If C is a subgraph of G, with a vertex v with 2 incoming edges, by definition of simple cycle for directed graphs, is C a simple cycle? I am trying to prove something about graphs... clearly a non-directed graph like this:
$a-b-c-d-a$ is a cycle, but what about when we talk about directed graphs,
$a \to b \to c \to d \to a$
is one, but is
$a \to b \leftarrow c \to d \to a$
one?
 A: As Wikipedia says:

A closed walk consists of a sequence of vertices starting and ending at the same vertex, with each two consecutive vertices in the sequence adjacent to each other in the graph. In a directed graph, each edge must be traversed by the walk consistently with its direction: the edge must be oriented from the earlier of two consecutive vertices to the later of the two vertices in the sequence. The choice of starting vertex is not important: traversing the same cyclic sequence of edges from different starting vertices produces the same closed walk.
A simple cycle may be defined either as a closed walk with no
  repetitions of vertices and edges allowed, other than the repetition
  of the starting and ending vertex, or as the set of edges in such a
  walk. The two definitions are equivalent in directed graphs, where
  simple cycles are also called directed cycles

Therefore we see that your example is not a simple cycle since you do not traverse the vertices in a same direction.
