Given a binary matrix mat[][] that represents the adjacency matrix representation of a graph, where mat[i][j] as 1 represents that there is an edge between vertices i and j and a vertex V, the task is to find the longest path from any node to the vertex X such that every vertex in the path occurs only once.
Examples:
Input: graph[][] = {{0, 1, 0, 0}, {1, 0, 1, 1}, {0, 1, 0, 0}, {0, 1, 0, 0}}, V = 2
Output: 2
Explanation:
The given graph is as follows:
0

1
/ \
2 3
The longest path ending at vertex 2 is 3 > 1 > 2. Therefore, the length of this path is 2.Input: graph[][] = {{0, 1, 1, 1}, {1, 0, 0, 0}, {1, 0, 0, 0}, {1, 0, 0, 0}}, V = 1
Output: 2
Approach: The given problem can be solved by performing DFS Traversal on the given graph from the source node as V and finding the maximum length of the path having the deepest node from the node V.
Follow the steps below to solve the problem:
 Initialize an adjacency list, say Adj[], from the given Graph representation in the matrix mat[][].
 Initialize an auxiliary vector, say visited[], to keep track of whether any vertex is visited or not.
 Initialize a variable, say distance as 0, to store the maximum length of the resultant path from any source node to the given vertex V.
 Perform DFS Traversal on the given graph from the node V and perform the following steps:
 Mark the current node V as visited, i.e., visited[V] = true.
 Update the value of distance to the maximum of distance and level.
 Traverse the adjacency list of the current source node V. If the child node is not same as the parent node and is not visited yet, then recursively perform DFS Traversal as dfs(child, Adj, visited, level + 1, distance).
 After completing the above steps, mark the current node V as unvisited to include the path if the cycle exists in the given graph.
 After completing the above steps, print the value of distance as the resultant maximum distance between any source node the given node V.
Below is the implementation of the above approach:
C++
// C++ program for the above approach #include <bits/stdc++.h> using namespace std; // Function to perform DFS Traversal from // source node to the deepest node and // update maximum distance to the deepest node void dfs( int src, vector< int > Adj[], vector< bool >& visited, int level, int & distance) { // Mark source as visited visited[src] = true ; // Update the maximum distance distance = max(distance, level); // Traverse the adjacency list // of the current source node for ( auto & child : Adj[src]) { // Recursively call // for the child node if (child != src and visited[child] == false ) { dfs(child, Adj, visited, level + 1, distance); } } // Backtracking step visited[src] = false ; } // Function to calculate maximum length // of the path ending at vertex V from // any source node int maximumLength(vector<vector< int > >& mat, int V) { // Stores the maximum length of // the path ending at vertex V int distance = 0; // Stores the size of the matrix int N = ( int )mat.size(); // Stores the adjacency list // of the given graph vector< int > Adj[N]; vector< bool > visited(N, false ); // Traverse the matrix to // create adjacency list for ( int i = 0; i < N; i++) { for ( int j = 0; j < N; j++) { if (mat[i][j] == 1) { Adj[i].push_back(j); } } } // Perform DFS Traversal to // update the maximum distance dfs(V, Adj, visited, 0, distance); return distance; } // Driver Code int main() { vector<vector< int > > mat = { { 0, 1, 0, 0 }, { 1, 0, 1, 1 }, { 0, 1, 0, 0 }, { 0, 1, 0, 0 } }; int V = 2; cout << maximumLength(mat, V); return 0; } 
Java
// Java program for the above approach import java.util.ArrayList; class GFG{ static int distance; // Function to perform DFS Traversal from source node to // the deepest node and update maximum distance to the // deepest node private static void dfs( int src, ArrayList<ArrayList<Integer>> Adj, ArrayList<Boolean> visited, int level) { // Mark source as visited visited.set(src, true ); // Update the maximum distance distance = Math.max(distance, level); // Traverase the adjacency list of the current // source node for ( int child : Adj.get(src)) { // Recursively call for the child node if ((child != src) && (visited.get(child) == false )) { dfs(child, Adj, visited, level + 1 ); } } // Backtracking step visited.set(src, false ); } // Function to calculate maximum length of the path // ending at vertex v from any source node private static int maximumLength( int [][] mat, int v) { // Stores the maximum length of the path ending at // vertex v distance = 0 ; // Stores the size of the matrix int N = mat[ 0 ].length; ArrayList<Boolean> visited = new ArrayList<Boolean>(); for ( int i = 0 ; i < N; i++) { visited.add( false ); } // Stores the adjacency list of the given graph ArrayList< ArrayList<Integer>> Adj = new ArrayList< ArrayList<Integer>>(N); for ( int i = 0 ; i < N; i++) { Adj.add( new ArrayList<Integer>()); } int i, j; // Traverse the matrix to create adjacency list for (i = 0 ; i < mat[ 0 ].length; i++) { for (j = 0 ; j < mat.length; j++) { if (mat[i][j] == 1 ) { Adj.get(i).add(j); } } } // Perform DFS Traversal to update // the maximum distance dfs(v, Adj, visited, 0 ); return distance; } // Driver code public static void main(String[] args) { int [][] mat = { { 0 , 1 , 0 , 0 }, { 1 , 0 , 1 , 1 }, { 0 , 1 , 0 , 0 }, { 0 , 1 , 0 , 0 } }; int v = 2 ; System.out.print(maximumLength(mat, v)); } } // This code is contributed by abhinavjain194 
Python3
# Python3 program for the above approach visited = [ False for i in range ( 4 )] # Function to perform DFS Traversal from # source node to the deepest node and # update maximum distance to the deepest node def dfs(src, Adj, level, distance): global visited # Mark source as visited visited[src] = True # Update the maximum distance distance = max (distance, level) # Traverse the adjacency list # of the current source node for child in Adj[src]: # Recursively call # for the child node if (child ! = src and visited[child] = = False ): dfs(child, Adj, level + 1 , distance) # Backtracking step visited[src] = False # Function to calculate maximum length # of the path ending at vertex V from # any source node def maximumLength(mat, V): # Stores the maximum length of # the path ending at vertex V distance = 0 # Stores the size of the matrix N = len (mat) # Stores the adjacency list # of the given graph Adj = [[] for i in range (N)] # Traverse the matrix to # create adjacency list for i in range (N): for j in range (N): if (mat[i][j] = = 1 ): Adj[i].append(j) # Perform DFS Traversal to # update the maximum distance dfs(V, Adj, 0 , distance) return distance + 2 # Driver Code if __name__ = = '__main__' : mat = [ [ 0 , 1 , 0 , 0 ], [ 1 , 0 , 1 , 1 ], [ 0 , 1 , 0 , 0 ], [ 0 , 1 , 0 , 0 ] ] V = 2 print (maximumLength(mat, V)) # This code is contributed by ipg2016107 
2
Time Complexity: O(N^{2})
Auxiliary Space: O(N)
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