# Find if there is a path between two vertices in a directed graph

Given a Directed Graph and two vertices in it, check whether there is a path from the first given vertex to second. For example, in the following graph, there is a path from vertex 1 to 3. As another example, there is no path from 3 to 0.

We can either use Breadth First Search (BFS) or Depth First Search (DFS) to find path between two vertices. Take the first vertex as source in BFS (or DFS), follow the standard BFS (or DFS). If we see the second vertex in our traversal, then return true. Else return false.

Following are C++,Java and Python codes that use BFS for finding reachability of second vertex from first vertex.

## C++

```// C++ program to check if there is exist a path between two vertices
// of a graph.
#include<iostream>
#include <list>
using namespace std;

// This class represents a directed graph using adjacency list
// representation
class Graph
{
int V;    // No. of vertices
public:
Graph(int V);  // Constructor
void addEdge(int v, int w); // function to add an edge to graph
bool isReachable(int s, int d);
};

Graph::Graph(int V)
{
this->V = V;
}

{
}

// A BFS based function to check whether d is reachable from s.
bool Graph::isReachable(int s, int d)
{
// Base case
if (s == d)
return true;

// Mark all the vertices as not visited
bool *visited = new bool[V];
for (int i = 0; i < V; i++)
visited[i] = false;

// Create a queue for BFS
list<int> queue;

// Mark the current node as visited and enqueue it
visited[s] = true;
queue.push_back(s);

// it will be used to get all adjacent vertices of a vertex
list<int>::iterator i;

while (!queue.empty())
{
// Dequeue a vertex from queue and print it
s = queue.front();
queue.pop_front();

// Get all adjacent vertices of the dequeued vertex s
// If a adjacent has not been visited, then mark it visited
// and enqueue it
{
// If this adjacent node is the destination node, then
// return true
if (*i == d)
return true;

// Else, continue to do BFS
if (!visited[*i])
{
visited[*i] = true;
queue.push_back(*i);
}
}
}

// If BFS is complete without visiting d
return false;
}

// Driver program to test methods of graph class
int main()
{
// Create a graph given in the above diagram
Graph g(4);

int u = 1, v = 3;
if(g.isReachable(u, v))
cout<< "\n There is a path from " << u << " to " << v;
else
cout<< "\n There is no path from " << u << " to " << v;

u = 3, v = 1;
if(g.isReachable(u, v))
cout<< "\n There is a path from " << u << " to " << v;
else
cout<< "\n There is no path from " << u << " to " << v;

return 0;
}
```

## Java

```// Java program to check if there is exist a path between two vertices
// of a graph.
import java.io.*;
import java.util.*;

// This class represents a directed graph using adjacency list
// representation
class Graph
{
private int V;   // No. of vertices

//Constructor
Graph(int v)
{
V = v;
for (int i=0; i<v; ++i)
}

//Function to add an edge into the graph

//prints BFS traversal from a given source s
Boolean isReachable(int s, int d)
{

// Mark all the vertices as not visited(By default set
// as false)
boolean visited[] = new boolean[V];

// Create a queue for BFS

// Mark the current node as visited and enqueue it
visited[s]=true;

// 'i' will be used to get all adjacent vertices of a vertex
Iterator<Integer> i;
while (queue.size()!=0)
{
// Dequeue a vertex from queue and print it
s = queue.poll();

int n;

// Get all adjacent vertices of the dequeued vertex s
// If a adjacent has not been visited, then mark it
// visited and enqueue it
while (i.hasNext())
{
n = i.next();

// If this adjacent node is the destination node,
// then return true
if (n==d)
return true;

// Else, continue to do BFS
if (!visited[n])
{
visited[n] = true;
}
}
}

// If BFS is complete without visited d
return false;
}

// Driver method
public static void main(String args[])
{
// Create a graph given in the above diagram
Graph g = new Graph(4);

int u = 1;
int v = 3;
if (g.isReachable(u, v))
System.out.println("There is a path from " + u +" to " + v);
else
System.out.println("There is no path from " + u +" to " + v);;

u = 3;
v = 1;
if (g.isReachable(u, v))
System.out.println("There is a path from " + u +" to " + v);
else
System.out.println("There is no path from " + u +" to " + v);;
}
}
// This code is contributed by Aakash Hasija
```

## Python

```# program to check if there is exist a path between two vertices
# of a graph

from collections import defaultdict

#This class represents a directed graph using adjacency list representation
class Graph:

def __init__(self,vertices):
self.V= vertices #No. of vertices
self.graph = defaultdict(list) # default dictionary to store graph

# function to add an edge to graph
self.graph[u].append(v)

# Use BFS to check path between s and d
def isReachable(self, s, d):
# Mark all the vertices as not visited
visited =[False]*(self.V)

# Create a queue for BFS
queue=[]

# Mark the source node as visited and enqueue it
queue.append(s)
visited[s] = True

while queue:

#Dequeue a vertex from queue
n = queue.pop(0)

# If this adjacent node is the destination node,
# then return true
if n == d:
return True

#  Else, continue to do BFS
for i in self.graph[n]:
if visited[i] == False:
queue.append(i)
visited[i] = True
# If BFS is complete without visited d
return False

# Create a graph given in the above diagram
g = Graph(4)

u =1; v = 3

if g.isReachable(u, v):
print("There is a path from %d to %d" % (u,v))
else :
print("There is no path from %d to %d" % (u,v))

u = 3; v = 1
if g.isReachable(u, v) :
print("There is a path from %d to %d" % (u,v))
else :
print("There is no path from %d to %d" % (u,v))

#This code is contributed by Neelam Yadav
```

Output:

``` There is a path from 1 to 3
There is no path from 3 to 1
```

As an exercise, try an extended version of the problem where the complete path between two vertices is also needed.

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