# Check if the given permutation is a valid DFS of graph

Given a graph with N nodes numbered from 1 to N and M edges and an array of numbers from 1 to N. Check if it is possible to obtain any permutation of array by applying DFS (Depth First Traversal) on given graph.

Prerequisites : DFS | Map in CPP

Examples :

```Input : N = 3, M = 2
Edges are:
1) 1-2
2) 2-3
P = {1, 2, 3}
Output : YES
Explanation :
Since there are edges between
1-2 and 2-3, therefore we can
have DFS in the order 1-2-3

Input : N = 3, M = 2
Edges are:
1) 1-2
2) 2-3
P = {1, 3, 2}
Output : NO
Explanation :
Since there is no edge between 1 and 3,
the DFS traversal is not possible
in the order of given permutation.
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Possible Permutations in which whole graph can be traversed are:
1) 1 2 3
2) 3 2 1

Approach : We assume that the input graph is represented as adjacency list. The idea is to first sort all adjacency lists according to input order, then traverse the given graph starting from first node in given given permutation. If we visit all vertices in same order, then given permutation is a valid DFS.

1. Store the indexes of each number in the given permutation in a Hash map.
2. Sort every adjacency list according to the indexes of permutation since there is need to maintain the order.
3. Perform the Depth First Traversal Search with source node as 1st number of given permutation.
4. Keep a counter variable and at every recursive call, check if the counter has reached the number of nodes, i.e. N and set the flag as 1. If the flag is 0 after complete DFS, answer is ‘NO’ otherwise ‘YES’

Below is the implementation of above approach :

 `// CPP program to check if given ` `// permutation can be obtained ` `// upon DFS traversal on given graph ` `#include ` `using` `namespace` `std; ` ` `  `// To track of DFS is valid or not. ` `bool` `flag = ``false``; ` ` `  `// HashMap to store the indexes ` `// of given permutation ` `map<``int``, ``int``> mp; ` ` `  `// Comparator function for sort ` `bool` `cmp(``int` `a, ``int` `b) ` `{ ` `    ``// Sort according ascending ` `    ``// order of indexes ` `    ``return` `mp[a] < mp[b]; ` `} ` ` `  `// Graph class represents a undirected ` `// using adjacency list representation ` `class` `Graph  ` `{ ` `    ``int` `V; ``// No. of vertices ` `    ``int` `counter; ``// Counter variable ` ` `  `public``: ` `    ``// Pointer to an array containing ` `    ``// adjacency lists ` `    ``list<``int``>* adj; ` ` `  `    ``Graph(``int` `V); ``// Constructor ` ` `  `    ``// function to add an edge to graph ` `    ``void` `addEdge(``int` `u, ``int` `v); ` ` `  `    ``// DFS traversal of the vertices ` `    ``// reachable from v ` `    ``void` `DFS(``int` `v, ``int` `Perm[]); ` `}; ` ` `  `Graph::Graph(``int` `V) ` `{ ` `    ``this``->V = V; ` `    ``this``->counter = 0; ` `    ``adj = ``new` `list<``int``>[V + 1]; ` `} ` ` `  `void` `Graph::addEdge(``int` `u, ``int` `v) ` `{ ` `    ``adj[u].push_back(v); ``// Add v to u’s list. ` `    ``adj[v].push_back(u); ``// Add u to v's list ` `} ` ` `  `// DFS traversal of the  ` `// vertices reachable from v. ` `void` `Graph::DFS(``int` `v, ``int` `Perm[]) ` `{ ` `    ``// Increment counter for  ` `    ``// every node being traversed ` `    ``counter++; ` ` `  `    ``// Check if counter has ` `    ``// reached number of vertices ` `    ``if` `(counter == V) { ` ` `  `        ``// Set flag to 1 ` `        ``flag = 1; ` `        ``return``; ` `    ``} ` ` `  `    ``// Recur for all vertices adjacent ` `    ``// to this vertices only if it  ` `    ``// lies in the given permutation ` `    ``list<``int``>::iterator i; ` `    ``for` `(i = adj[v].begin();  ` `         ``i != adj[v].end(); i++)  ` `    ``{ ` ` `  `        ``// if the current node equals to ` `        ``// current element of permutation ` `        ``if` `(*i == Perm[counter]) ` `            ``DFS(*i, Perm); ` `    ``} ` `} ` ` `  `// Returns true if P[] is a valid DFS of given ` `// graph. In other words P[] can be obtained by ` `// doing a DFS of the graph. ` `bool` `checkPermutation(``int` `N, ``int` `M,  ` `      ``vector > V, ``int` `P[]) ` `{ ` `    ``// Create the required graph with  ` `    ``// N vertices and M edges ` `    ``Graph G(N); ` ` `  `    ``// Add Edges to Graph G ` `    ``for` `(``int` `i = 0; i < M; i++)  ` `        ``G.addEdge(V[i].first, V[i].second);     ` ` `  `    ``for` `(``int` `i = 0; i < N; i++) ` `        ``mp[P[i]] = i; ` ` `  `    ``// Sort every adjacency  ` `    ``// list according to HashMap ` `    ``for` `(``int` `i = 1; i <= N; i++) ` `        ``G.adj[i].sort(cmp); ` ` `  `    ``// Call DFS with source node as P[0] ` `    ``G.DFS(P[0], P); ` ` `  `    ``// If Flag has been set to 1, means ` `    ``// given permutation is obtained  ` `    ``// by DFS on given graph ` `    ``return` `flag; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``// Number of vertices and number of edges ` `    ``int` `N = 3, M = 2; ` ` `  `    ``// Vector of pair to store edges ` `    ``vector > V; ` ` `  `    ``V.push_back(make_pair(1, 2)); ` `    ``V.push_back(make_pair(2, 3)); ` ` `  `    ``int` `P[] = { 1, 2, 3 }; ` ` `  `    ``// Return the answer ` `    ``if` `(checkPermutation(N, M, V, P)) ` `        ``cout << ``"YES"` `<< endl; ` `    ``else` `        ``cout << ``"NO"` `<< endl; ` ` `  `    ``return` `0; ` `} `

Output:

```YES
```

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