Given a Directed Graph and two vertices in it, check whether there is a path from the first given vertex to second.**Example:**

Consider the following Graph:

Input :(u, v) = (1, 3)Output:YesExplanation:

There is a path from 1 to 3, 1 -> 2 -> 3

Input :(u, v) = (3, 6)Output:NoExplanation:

There is no path from 3 to 6

A BFS or DFS based solution of this problem is discuss here.**Approach :** Here we will discuss a Dynamic Programming based solution using Floyd Warshall Algorithm.

- Create a boolean 2D matrix
**mat**where**mat[i][j]**will be true if there is a path from vertex**i**to**j**. - For every starting vertex
**i**and ending vertex**j**iterate over all intermediate vertex**k**and do check if there is a path for**i**to**j**through**k**then mark**mat[i][j]**as true. - Finally check if
**mat[u][v]**is true then return true else return false.

Below is the implementation of the above approach :

## C++

`// C++ program to find if there is a` `// path between two vertices in a` `// directed graph using Dynamic Programming` `#include <bits/stdc++.h>` `using` `namespace` `std;` `#define X 6` `#define Z 2` `// function to find if there is a` `// path between two vertices in a` `// directed graph` `bool` `existPath(` `int` `V, ` `int` `edges[X][Z],` ` ` `int` `u, ` `int` `v)` `{` ` ` `// dp matrix` ` ` `bool` `mat[V][V];` ` ` `memset` `(mat, ` `false` `, ` `sizeof` `(mat));` ` ` `// set dp[i][j]=true if there is` ` ` `// edge between i to j` ` ` `for` `(` `int` `i = 0; i < X; i++)` ` ` `mat[edges[i][0]][edges[i][1]] = ` `true` `;` ` ` `// check for all intermediate vertex` ` ` `for` `(` `int` `k = 0; k < V; k++) {` ` ` `for` `(` `int` `i = 0; i < V; i++) {` ` ` `for` `(` `int` `j = 0; j < V; j++) {` ` ` `mat[i][j] = mat[i][j]` ` ` `|| mat[i][k]` ` ` `&& mat[k][j];` ` ` `}` ` ` `}` ` ` `}` ` ` `// if vertex is invalid` ` ` `if` `(u >= V || v >= V) {` ` ` `return` `false` `;` ` ` `}` ` ` `// if there is a path` ` ` `if` `(mat[u][v])` ` ` `return` `true` `;` ` ` `return` `false` `;` `}` `// Driver function` `int` `main()` `{` ` ` `int` `V = 4;` ` ` `int` `edges[X][Z]` ` ` `= { { 0, 2 }, { 0, 1 },` ` ` `{ 1, 2 }, { 2, 3 },` ` ` `{ 2, 0 }, { 3, 3 } };` ` ` `int` `u = 1, v = 3;` ` ` `if` `(existPath(V, edges, u, v))` ` ` `cout << ` `"Yes\n"` `;` ` ` `else` ` ` `cout << ` `"No\n"` `;` ` ` `return` `0;` `}` |

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## Java

`// Java program to find if there is a path ` `// between two vertices in a directed graph` `// using Dynamic Programming` `import` `java.util.*;` `class` `GFG{` ` ` `static` `final` `int` `X = ` `6` `;` `static` `final` `int` `Z = ` `2` `;` `// Function to find if there is a` `// path between two vertices in a` `// directed graph` `static` `boolean` `existPath(` `int` `V, ` `int` `edges[][],` ` ` `int` `u, ` `int` `v)` `{` ` ` ` ` `// mat matrix` ` ` `boolean` `[][]mat = ` `new` `boolean` `[V][V];` ` ` `// set mat[i][j]=true if there is` ` ` `// edge between i to j` ` ` `for` `(` `int` `i = ` `0` `; i < X; i++)` ` ` `mat[edges[i][` `0` `]][edges[i][` `1` `]] = ` `true` `;` ` ` `// Check for all intermediate vertex` ` ` `for` `(` `int` `k = ` `0` `; k < V; k++) ` ` ` `{` ` ` `for` `(` `int` `i = ` `0` `; i < V; i++) ` ` ` `{` ` ` `for` `(` `int` `j = ` `0` `; j < V; j++)` ` ` `{` ` ` `mat[i][j] = mat[i][j] || ` ` ` `mat[i][k] &&` ` ` `mat[k][j];` ` ` `}` ` ` `}` ` ` `}` ` ` `// If vertex is invalid` ` ` `if` `(u >= V || v >= V)` ` ` `{` ` ` `return` `false` `;` ` ` `}` ` ` `// If there is a path` ` ` `if` `(mat[u][v])` ` ` `return` `true` `;` ` ` `return` `false` `;` `}` `// Driver code` `public` `static` `void` `main(String[] args)` `{` ` ` `int` `V = ` `4` `;` ` ` `int` `edges[][] = { { ` `0` `, ` `2` `}, { ` `0` `, ` `1` `},` ` ` `{ ` `1` `, ` `2` `}, { ` `2` `, ` `3` `},` ` ` `{ ` `2` `, ` `0` `}, { ` `3` `, ` `3` `} };` ` ` `int` `u = ` `1` `, v = ` `3` `;` ` ` `if` `(existPath(V, edges, u, v))` ` ` `System.out.print(` `"Yes\n"` `);` ` ` `else` ` ` `System.out.print(` `"No\n"` `);` `}` `}` `// This code is contributed by Princi Singh` |

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## Python3

`# Python3 program to find if there` `# is a path between two vertices in a` `# directed graph using Dynamic Programming` `X ` `=` `6` `Z ` `=` `2` ` ` `# Function to find if there is a` `# path between two vertices in a` `# directed graph` `def` `existPath(V, edges, u, v):` ` ` ` ` `# dp matrix` ` ` `mat ` `=` `[[` `False` `for` `i ` `in` `range` `(V)] ` ` ` `for` `j ` `in` `range` `(V)]` ` ` ` ` `# Set dp[i][j]=true if there is` ` ` `# edge between i to j` ` ` `for` `i ` `in` `range` `(X):` ` ` `mat[edges[i][` `0` `]][edges[i][` `1` `]] ` `=` `True` ` ` ` ` `# Check for all intermediate vertex` ` ` `for` `k ` `in` `range` `(V):` ` ` `for` `i ` `in` `range` `(V):` ` ` `for` `j ` `in` `range` `(V):` ` ` `mat[i][j] ` `=` `(mat[i][j] ` `or` ` ` `mat[i][k] ` `and` ` ` `mat[k][j])` ` ` ` ` `# If vertex is invalid` ` ` `if` `(u >` `=` `V ` `or` `v >` `=` `V):` ` ` `return` `False` ` ` ` ` `# If there is a path` ` ` `if` `(mat[u][v]):` ` ` `return` `True` ` ` ` ` `return` `False` `# Driver code ` `V ` `=` `4` `edges ` `=` `[ [ ` `0` `, ` `2` `], [ ` `0` `, ` `1` `],` ` ` `[ ` `1` `, ` `2` `], [ ` `2` `, ` `3` `],` ` ` `[ ` `2` `, ` `0` `], [ ` `3` `, ` `3` `] ]` ` ` `u, v ` `=` `1` `, ` `3` `if` `(existPath(V, edges, u, v)):` ` ` `print` `(` `"Yes"` `)` `else` `:` ` ` `print` `(` `"No"` `)` `# This code is contributed by divyeshrabadiya07` |

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## C#

`// C# program to find if there is a path ` `// between two vertices in a directed graph` `// using Dynamic Programming` `using` `System;` `class` `GFG{` ` ` `static` `readonly` `int` `X = 6;` `static` `readonly` `int` `Z = 2;` `// Function to find if there is a` `// path between two vertices in a` `// directed graph` `static` `bool` `existPath(` `int` `V, ` `int` `[,]edges,` ` ` `int` `u, ` `int` `v)` `{` ` ` ` ` `// mat matrix` ` ` `bool` `[,]mat = ` `new` `bool` `[V, V];` ` ` `// set mat[i,j]=true if there is` ` ` `// edge between i to j` ` ` `for` `(` `int` `i = 0; i < X; i++)` ` ` `mat[edges[i, 0], edges[i, 1]] = ` `true` `;` ` ` `// Check for all intermediate vertex` ` ` `for` `(` `int` `k = 0; k < V; k++) ` ` ` `{` ` ` `for` `(` `int` `i = 0; i < V; i++) ` ` ` `{` ` ` `for` `(` `int` `j = 0; j < V; j++)` ` ` `{` ` ` `mat[i, j] = mat[i, j] || ` ` ` `mat[i, k] &&` ` ` `mat[k, j];` ` ` `}` ` ` `}` ` ` `}` ` ` `// If vertex is invalid` ` ` `if` `(u >= V || v >= V)` ` ` `{` ` ` `return` `false` `;` ` ` `}` ` ` `// If there is a path` ` ` `if` `(mat[u, v])` ` ` `return` `true` `;` ` ` `return` `false` `;` `}` `// Driver code` `public` `static` `void` `Main(String[] args)` `{` ` ` `int` `V = 4;` ` ` `int` `[,]edges = { { 0, 2 }, { 0, 1 },` ` ` `{ 1, 2 }, { 2, 3 },` ` ` `{ 2, 0 }, { 3, 3 } };` ` ` `int` `u = 1, v = 3;` ` ` `if` `(existPath(V, edges, u, v))` ` ` `Console.Write(` `"Yes\n"` `);` ` ` `else` ` ` `Console.Write(` `"No\n"` `);` `}` `}` `// This code is contributed by sapnasingh4991` |

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**Output:**

Yes

**Time Complexity :** O ( V^{ 3})**Auxiliary Space :** O ( V^{ 2})

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