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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace AtCoder2 {
	class Program {
		static void Main(string[] args) {

	
			var inputArray = retreiveLongArray();
			var inputCount = 2;

			// 不正な入力は処理しない
			if (inputArray.Length != inputCount) {
				Console.WriteLine("invalid input");
				return;
			}
			var N = (int)inputArray[0];
			var M = (int)inputArray[1];

			var hens = new List<Tuple<int, int>>();

			for (int i = 0; i < M; i++) {
				var inpArray = retreiveLongArray();
				var hen = new Tuple<int, int>((int)inpArray[0], (int)inpArray[1]);
				hens.Add(hen);
			}

			_map = new Dictionary<int, List<int>>();
			for (int i = 1; i <= N; i++) {
				_map.Add(i, new List<int>());
			}

			for (int i = 0; i < M; i++) {

				var hen = hens[i];

				_map[hen.Item1].Add(hen.Item2);
				_map[hen.Item2].Add(hen.Item1);

			}

			_isToota = new Dictionary<int, bool>();
			for (int i = 1; i <= N; i++) {
				_isToota.Add(i, false);
			}

			_hitohude = 0;
			//最初は1からスタートする
			dFS(1,N);

			//答えの出力
			var ans = _hitohude;
			Console.WriteLine(ans);

		}

		//フィールドの定義
		private static Dictionary<int, List<int>> _map;
		private static Dictionary<int, bool> _isToota;
		private static int _hitohude;


		private static void dFS(int current,int choutenNum) {
			_isToota[current] = true;

			var tootta = _isToota.Where(x => x.Value).Count();
			if (tootta == choutenNum) {
				_hitohude++;
			}

			var nexts = _map[current];

			foreach (var next in nexts) {
				if (!_isToota[next]) {
					dFS(next,choutenNum);
				}
			}
			_isToota[current] = false;
			return;
		}

		/// <summary>
		/// スペースで区切られた文字列配列の入力を受け取る
		/// </summary>
		/// <returns></returns>
		private static string[] retreiveStringArray() {
			var inputArray = Console.ReadLine().Split(' ');
			return inputArray;
		}

		/// <summary>
		/// スペースで区切られた整数配列の入力を受け取る
		/// </summary>
		/// <returns></returns>
		private static long[] retreiveLongArray() {
			var inputArray = Console.ReadLine().Split(' ');
			var longArray = inputArray.Select(s => long.Parse(s));
			return longArray.ToArray();
		}
	}
}