# Check if vertex X lies in subgraph of vertex Y for the given Graph

Given an undirected graph and two vertices X and Y, our task is to check whether the vertex X lies in the subgraph of the vertex Y.
Examples:

Input: X = 2, Y = 3
Output: No
Explanation:
Subgraph of a vertex Y = 3 is set of all the vertex which lies below Y and are reachable by Y. Here subgraph of 3 contains {6} not 2.
Input: X = 6, Y = 1
Output: Yes
Explanation:
Subgraph of 1 contain {2, 3, 4, 5, 6} so 6 lies in subgraph of 1.

Approach: The idea is to use Depth First Search(DFS). Initialize two arrays in and out for maintaining the start time of traversing a vertex and end time to mark until the vertex traversed. If the starting time of the second vertex is less than the starting time of the first vertex and the ending time of the first vertex is less than that of the second vertex then return true else return false.
Below is the implementation of the above approach:

## C++

 `// C++ implementation to check if vertex X  ` `// lies in subgraph of vertex Y  ` `// for the given graph  ` `#include   ` `using` `namespace` `std;  ` `int` `cnt = 1;  ` ` `  `// Function ot perform dfs  ` `void` `dfs(vector<``int``> v[], ``int` `in[],  ` `        ``int` `out[], ``int` `visited[], ``int` `i)  ` `{  ` `    ``// Mark visited of vertex i  ` `    ``visited[i] = 1;  ` ` `  `    ``// Update starting time  ` `    ``// of vertex i  ` `    ``in[i] = cnt;  ` ` `  `    ``// Increment the cnt  ` `    ``cnt++;  ` ` `  `    ``for` `(``auto` `x : v[i]) {  ` `        ``// Check if not visited  ` `        ``// call dfs from x  ` `        ``if` `(!visited[x])  ` `            ``dfs(v, in, out, visited, x);  ` `    ``}  ` ` `  `    ``// Update ending time  ` `    ``// of vertex i  ` `    ``out[i] = cnt;  ` ` `  `    ``// Increment the cnt  ` `    ``cnt++;  ` `}  ` ` `  `// Function to add edges in graph  ` `void` `addedge(vector<``int``> v[], ``int` `x, ``int` `y)  ` `{  ` `    ``v[x].push_back(y);  ` `    ``v[y].push_back(x);  ` `}  ` ` `  `// Function to check if vertex X  ` `// lies in subgraph of vertex Y  ` `// for the given graph  ` `bool` `is_subtree(vector<``int``> v[], ``int` `n,  ` `                ``int` `m, ``int` `x, ``int` `y)  ` `{  ` `    ``// Arrays for starting time,  ` `    ``// ending time and to check  ` `    ``// for visited respectively  ` `    ``int` `in[n + 1], out[n + 1], visited[n + 1];  ` ` `  `    ``// Mark all vertices starting time,  ` `    ``// ending time and visited as zero  ` `    ``for` `(``int` `i = 1; i <= n; i++) {  ` `        ``in[i] = 0;  ` `        ``out[i] = 0;  ` `        ``visited[i] = 0;  ` `    ``}  ` ` `  `    ``// Check if y comes before x  ` `    ``// and leaves after x then x lies  ` `    ``// in the subgraph of y  ` `    ``// call dfs from any vertex,  ` `    ``// here we have called from 1  ` `    ``dfs(v, in, out, visited, 1);  ` `    ``if` `(in[y] < in[x] && out[y] > out[x])  ` `        ``return` `true``;  ` ` `  `    ``else` `        ``return` `false``;  ` `}  ` ` `  `// Driver code  ` `int` `main()  ` `{  ` `    ``// n number of vertices  ` `    ``// m number of edges  ` `    ``int` `n = 6, m = 5;  ` ` `  `    ``// Create a graph given  ` `    ``// in the above diagram  ` `    ``vector<``int``> v[n + 1];  ` `    ``addedge(v, 1, 2);  ` `    ``addedge(v, 1, 3);  ` `    ``addedge(v, 2, 4);  ` `    ``addedge(v, 1, 5);  ` `    ``addedge(v, 3, 6);  ` ` `  `    ``int` `x = 6, y = 1;  ` `    ``if` `(is_subtree(v, n, m, x, y))  ` `        ``cout << ``"Yes"``;  ` `    ``else` `        ``cout << ``"No"``;  ` ` `  `    ``return` `0;  ` `}  `

## Java

 `// Java implementation to check if vertex X  ` `// lies in subgraph of vertex Y  ` `// for the given graph  ` `import` `java.util.*; ` ` `  `class` `GFG{  ` `     `  `static` `int` `cnt = ``1``;  ` ` `  `// Function ot perform dfs  ` `static` `void` `dfs(Vector v[], ``int` `in[],  ` `                ``int` `out[], ``int` `visited[], ``int` `i)  ` `{  ` `     `  `    ``// Mark visited of vertex i  ` `    ``visited[i] = ``1``;  ` ` `  `    ``// Update starting time  ` `    ``// of vertex i  ` `    ``in[i] = cnt;  ` ` `  `    ``// Increment the cnt  ` `    ``cnt++;  ` ` `  `    ``for``(``int` `x : v[i])  ` `    ``{ ` `         `  `        ``// Check if not visited  ` `        ``// call dfs from x  ` `        ``if` `(visited[x] == ``0``)  ` `            ``dfs(v, in, out, visited, x);  ` `    ``}  ` ` `  `    ``// Update ending time  ` `    ``// of vertex i  ` `    ``out[i] = cnt;  ` ` `  `    ``// Increment the cnt  ` `    ``cnt++;  ` `}  ` ` `  `// Function to add edges in graph  ` `static` `void` `addedge(Vector v[],  ` `                    ``int` `x, ``int` `y)  ` `{  ` `    ``v[x].add(y);  ` `    ``v[y].add(x);  ` `}  ` ` `  `// Function to check if vertex X  ` `// lies in subgraph of vertex Y  ` `// for the given graph  ` `static` `boolean` `is_subtree(Vector v[], ` `                          ``int` `n, ``int` `m, ``int` `x, ` `                          ``int` `y)  ` `{  ` `     `  `    ``// Arrays for starting time,  ` `    ``// ending time and to check  ` `    ``// for visited respectively  ` `    ``int` `[]in = ``new` `int``[n + ``1``]; ` `    ``int` `[]out = ``new` `int``[n + ``1``]; ` `    ``int` `[]visited = ``new` `int``[n + ``1``];  ` ` `  `    ``// Mark all vertices starting time,  ` `    ``// ending time and visited as zero  ` `    ``for``(``int` `i = ``1``; i <= n; i++)  ` `    ``{  ` `        ``in[i] = ``0``;  ` `        ``out[i] = ``0``;  ` `        ``visited[i] = ``0``;  ` `    ``}  ` ` `  `    ``// Check if y comes before x  ` `    ``// and leaves after x then x lies  ` `    ``// in the subgraph of y  ` `    ``// call dfs from any vertex,  ` `    ``// here we have called from 1  ` `    ``dfs(v, in, out, visited, ``1``);  ` `     `  `    ``if` `(in[y] < in[x] && out[y] > out[x])  ` `        ``return` `true``;  ` `    ``else` `        ``return` `false``;  ` `}  ` ` `  `// Driver code  ` `public` `static` `void` `main(String[] args)  ` `{  ` `     `  `    ``// n number of vertices  ` `    ``// m number of edges  ` `    ``int` `n = ``6``, m = ``5``;  ` ` `  `    ``// Create a graph given  ` `    ``// in the above diagram  ` `    ``@SuppressWarnings``(``"unchecked"``) ` `    ``Vector []v = ``new` `Vector[n + ``1``];  ` `    ``for``(``int` `i = ``0``; i < v.length; i++) ` `        ``v[i] = ``new` `Vector(); ` `         `  `    ``addedge(v, ``1``, ``2``);  ` `    ``addedge(v, ``1``, ``3``);  ` `    ``addedge(v, ``2``, ``4``);  ` `    ``addedge(v, ``1``, ``5``);  ` `    ``addedge(v, ``3``, ``6``);  ` ` `  `    ``int` `x = ``6``, y = ``1``;  ` `    ``if` `(is_subtree(v, n, m, x, y))  ` `        ``System.out.print(``"Yes"``);  ` `    ``else` `        ``System.out.print(``"No"``);  ` `}  ` `}  ` ` `  `// This code is contributed by PrinciRaj1992`

## Python3

 `# Python3 implementation to check if  ` `# vertex X lies in subgraph of  ` `# vertex Y for the given graph  ` `cnt ``=` `1` ` `  `# Function to perform dfs ` `def` `dfs(v, in_, out, visited, i): ` `     `  `    ``global` `cnt ` `     `  `    ``# Mark visited of vertex i ` `    ``visited[i] ``=` `1` `     `  `    ``# Update starting time  ` `    ``# of vertex i  ` `    ``in_[i] ``=` `cnt ` `     `  `    ``# Increment the cnt ` `    ``cnt ``+``=` `1` `     `  `    ``# Check if not visited  ` `    ``# call dfs from x  ` `    ``for` `x ``in` `v[i]: ` `        ``if` `not` `visited[x]: ` `            ``dfs(v, in_, out, visited, x) ` `             `  `    ``# Update ending time  ` `    ``# of vertex i  ` `    ``out[i] ``=` `cnt ` `     `  `    ``# Increment the cnt  ` `    ``cnt ``+``=` `1` `     `  `# Function to add edges in graph ` `def` `addedge(v, x, y): ` `     `  `    ``v[x].append(y) ` `    ``v[y].append(x) ` `         `  `# Function to check if vertex X  ` `# lies in subgraph of vertex Y  ` `# for the given graph ` `def` `is_subtree(v, n, m, x, y): ` `     `  `    ``# Arrays for starting time,  ` `    ``# ending time and to check  ` `    ``# for visited respectively  ` `     `  `    ``# Mark all vertices starting time,  ` `    ``# ending time and visited as zero ` `    ``in_ ``=` `[``0``] ``*` `(n ``+` `1``) ` `    ``out ``=` `[``0``] ``*` `(n ``+` `1``) ` `    ``visited ``=` `[``0``] ``*` `(n ``+` `1``) ` `     `  `    ``# Check if y comes before x  ` `    ``# and leaves after x then x lies  ` `    ``# in the subgraph of y  ` `    ``# call dfs from any vertex,  ` `    ``# here we have called from 1  ` `    ``dfs(v, in_, out, visited, ``1``) ` `    ``if` `in_[y] < in_[x] ``and` `out[y] > out[x]: ` `        ``return` `True` `    ``else``: ` `        ``return` `False` `     `  `# Driver code ` ` `  `# n number of vertices  ` `# m number of edges  ` `n, m ``=` `6``, ``5` ` `  `# Create a graph given  ` `# in the above diagram  ` `v ``=` `[] ` `for` `i ``in` `range``(n ``+` `1``): ` `    ``v.append([]) ` ` `  `addedge(v, ``1``, ``2``) ` `addedge(v, ``1``, ``3``) ` `addedge(v, ``2``, ``4``) ` `addedge(v, ``1``, ``5``) ` `addedge(v, ``3``, ``6``) ` ` `  `x, y ``=` `6``, ``1` ` `  `if` `is_subtree(v, n, m, x, y): ` `    ``print``(``"Yes"``) ` `else``: ` `    ``print``(``"No"``) ` ` `  `# This code is contributed by Stuti Pathak `

Output:

```Yes
```

Time Complexity: O(V + E)
Space Complexity: O(3*N)

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