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 Python (3.4.3) (judge verdict: Wrong Answer)

# https://atcoder.jp/contests/abc054/tasks/abc054_c

import copy

# 頂点, 辺の数
N, M = map(int, input().split())

# グラフのリスト
# 各頂点からどこの頂点に向かう辺があるかどうかという情報
Graph_list = [[] for i in range(N)]

# 入力
for i in range(M):
    a, b = map(int, input().split())
    a += -1
    b += -1

    # 無向グラフなのでa, b両方
    Graph_list[a].append(b)
    Graph_list[b].append(a)

ans_count = 0
throwgh_point_list = [-1 for i in range(N)]

def dfs(i, current, throwgh_point_list, ans_point_list):   
    # 頂点をN個通ったら終了処理
    if i >= N:
        # すでにカウント済の経路である場合
        if throwgh_point_list in ans_point_list:
            return

        # すでにカウント済ではないかつ全ての点に到達している場合
        # 順不同で比較
        if set(throwgh_point_list) == set(list(range(N))):
            global ans_count
            ans_count += 1
            ans_point_list.append(copy.copy(throwgh_point_list))
            print(throwgh_point_list)
        return

    for j in range(len(Graph_list[current])):
        # 現在地を書き込む
        throwgh_point_list[i] = current

        # 次の点に移動する        
        dfs(i + 1, Graph_list[current][j], throwgh_point_list, ans_point_list)

dfs(0, 0, throwgh_point_list, [])
print(ans_count)