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 list<int> *adj; // Pointer to an array containing adjacency lists 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; adj = new list<int>[V]; } void Graph::addEdge(int v, int w) { adj[v].push_back(w); // Add w to v’s list. } // 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 for (i = adj[s].begin(); i != adj[s].end(); ++i) { // 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); g.addEdge(0, 1); g.addEdge(0, 2); g.addEdge(1, 2); g.addEdge(2, 0); g.addEdge(2, 3); g.addEdge(3, 3); 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.*; import java.util.LinkedList; // This class represents a directed graph using adjacency list // representation class Graph { private int V; // No. of vertices private LinkedList<Integer> adj[]; //Adjacency List //Constructor Graph(int v) { V = v; adj = new LinkedList[v]; for (int i=0; i<v; ++i) adj[i] = new LinkedList(); } //Function to add an edge into the graph void addEdge(int v,int w) { adj[v].add(w); } //prints BFS traversal from a given source s Boolean isReachable(int s, int d) { LinkedList<Integer>temp; // Mark all the vertices as not visited(By default set // as false) boolean visited[] = new boolean[V]; // Create a queue for BFS LinkedList<Integer> queue = new LinkedList<Integer>(); // Mark the current node as visited and enqueue it visited[s]=true; queue.add(s); // '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; i = adj[s].listIterator(); // 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; queue.add(n); } } } // 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); g.addEdge(0, 1); g.addEdge(0, 2); g.addEdge(1, 2); g.addEdge(2, 0); g.addEdge(2, 3); g.addEdge(3, 3); 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 def addEdge(self,u,v): 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) g.addEdge(0, 1) g.addEdge(0, 2) g.addEdge(1, 2) g.addEdge(2, 0) g.addEdge(2, 3) g.addEdge(3, 3) 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.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.