### Checking Presence of Cycle in Directed Graph using DFS

Here, we will be discussing an algorithm to detect the presence of the cycle in a directed graph.

Prerequisites
Example:
Below we have two directed graphs out of which in (b) we have a path from each node to another node were as in (a) we can't move from 2 to 3.

In graph (b) we have cycles whereas in a graph (a) don't have a cycle.

Algorithm to detect the presence of a cycle.
To find the presence of a cycle we will use colouring technique. Let us say,

• Color 0: Node is not been visited ( White Node )
• Color 1: Node is currently is in processing mode ( Gray Node )
• Color 2: Node and all it's adjacent are processed. ( Black Node )
The cycle could be found when the node currently getting visited is already in processing ie. color =1. This means that we have a different path to previously visited node and hence there is cycle.

Simulation of DFS in (a):
Now let us simulate the DFS algorithm in the graph (a) starting with node 1.
• Initially, each node has Color = 0, when we start DFS(1) then it calls DFS(2) from 2 we cannot call any other node so it is processed.
• Then 1 is processed. After which we start DFS(3) which calls DFS(4). Now from 4, we can visit 2 but it is already visited, so node 4 is also processed now, and so 3 is processed.
In this, we have not got any cycle.

Simulation of DFS in (b):
Now let us simulate the DFS algorithm in the graph (b) starting with node 1.
• Let us call DFS(1) which in turn calls DFS(3) which in turn calls DFS(4) which in turn calls DFS(2) which in turn calls DFS(1) but 1 is already visited and but not yet processed so we found a cycle.
Implementation in C++

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