**Objective**: Given a graph, source vertex and destination vertex. Write an algorithm to count all possible paths between source and destination.

Condition: Graph does not contain any cycle.

This problem also known as “paths between two nodes”

**Example**:

**Approach**: Use Depth First Search

- Use DFS but we cannot use visited [] to keep track of visited vertices since we need to explore all the paths. visited [] is used avoid going into cycles during iteration. (That is why we have a condition in this problem that graph does not contain cycle)
- Start from the source vertex and make a recursive call to all it adjacent vertices.
- During recursive call, if reach the destination vertex, increment the result (no of paths).
- See the code for more understanding.

**Code**:

**Output**:

No of paths between source: 0 and destination: 5 are: 3

You can even do with cycles. Set visited[start] = false when returning

private static void countAllPaths(int start, int end, boolean[] visited) {

visited[start] = true;

LinkedList adjacentList = graphList[start];

for (int i = 0; i < adjacentList.size(); i++) {

int destination = adjacentList.get(i);

if (destination != end && visited[destination] == false) {

countAllPaths(destination, end, visited);

} else if (destination == end) {

pathCount++;

}

}

//Remove to start again

visited[start] = false;

}