A cycle in a digraph is a sequence of vertices $v_1, v_2, \dots, v_{k}$ such that $(v_{i}, v_{i+1}) \in E$ for all $0 \leq i \leq k-1$ and $(v_k, v_1) \in E$. Here is an example of a directed graph with a cycle:

The number $k$ is the length of the cycle. We explicitly allow cycles of length 2:

and even of length $1$, also called self-loops.

Here is an example of a digraph without a cycle:

We can use depth-first search to detect cycles. When we explore a new edge $(u,v)$ and $v$ has already been visited, this can be a cycle but does not have to. It is a cycle of $v$ is still on the stack. Thus, we need an additional boolean array is_on_stack that lets us quickly look up whether a vertex is on the stack. When the depth-first search with starting point $s$ returns without having found a cycle, we cannot yet conclude that $G$ is acyclic; it might be that no cycle is reachable from $s$; we need to proceed to the next unvisited vertex $s'$ and now start a depth-first search from $s'$. Thus, we need to have yet another for loop to wrap our depth-first search in.

7 9 
1 2 
2 3 
3 4 
3 7 
4 6 
4 7 
5 2 
5 3 
6 1                                                                                 

has a cycle

8 10
1 2 
1 5 
1 6
2 3 
3 4 
3 7
4 6 
5 6 
8 2 
8 3  

is acyclic