Problem:
Score : 300 points

Problem Statement

You are given an undirected unweighted graph with N vertices and M edges that contains neither self-loops nor double edges.

Here, a self-loop is an edge where a_i = b_i (1≤i≤M), and double edges are two edges where (a_i,b_i)=(a_j,b_j) or (a_i,b_i)=(b_j,a_j) (1≤i<j≤M).

How many different paths start from vertex 1 and visit all the vertices exactly once?

Here, the endpoints of a path are considered visited.

For example, let us assume that the following undirected graph shown in Figure 1 is given.

Figure 1: an example of an undirected graph

The following path shown in Figure 2 satisfies the condition.

Figure 2: an example of a path that satisfies the condition

However, the following path shown in Figure 3 does not satisfy the condition, because it does not visit all the vertices.

Figure 3: an example of a path that does not satisfy the condition

Neither the following path shown in Figure 4, because it does not start from vertex 1.

Figure 4: another example of a path that does not satisfy the condition

Constraints

2≦N≦8

0≦M≦N(N-1)/2

1≦a_i<b_i≦N

The given graph contains neither self-loops nor double edges.

Input

The input is given from Standard Input in the following format:

N M
a_1 b_1
a_2 b_2
:
a_M b_M

Output

Print the number of the different paths that start from vertex 1 and visit all the vertices exactly once.

Sample Input 1

3 3
1 2
1 3
2 3

Sample Output 1

2

The given graph is shown in the following figure:

The following two paths satisfy the condition:

Sample Input 2

7 7
1 3
2 7
3 4
4 5
4 6
5 6
6 7

Sample Output 2

1

This test case is the same as the one described in the problem statement.

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <vector>

using namespace std;

int N;

int dfs(vector<int>* edges, vector<int> S, int node){
    if(S.size() == N) return 1;
    bool flag;
    int count = 0;
    for(int p : edges[node]){
        flag = false;
        for(int s : S){
            if(p == s){
                flag=true;
                break;
            }
        }
        // すでに辿ってきた
        if(flag) continue;
        else{
            S.push_back(p);
            count += dfs(edges, S, p);
            S.pop_back();
        }
    }
    return count;
}

int main()
{
    int M;
    cin >> N >> M;
    vector<int> edges[N];
    int a, b;
    for(int i=0;i<N;++i){
        cin >> a >> b;
        --a;
        --b;
        edges[a].push_back(b);
        edges[b].push_back(a);
    }

    int ans = dfs(edges, vector<int>(1), 0);

    cout << ans << endl;
}