（生成树计数）Join

http://acm.timus.ru/problem.aspx?space=1&num=1627

将给出的图跑出Kirchhoff矩阵C[G]即可
https://blog.youkuaiyun.com/weixin_40588429/article/details/87646662

#include <iostream>
#include <algorithm>
#include <cstdlib>
#include <cstring>
#include <vector>
#include <string>
#include <cmath>
using namespace std;
typedef long long ll;

const int maxn = 10 + 5;
const int mod = 1e9;
char g[maxn][maxn];
int id[maxn][maxn], dir[4][2] = {{-1, 0}, {1, 0}, {0, 1}, {0, -1}};
ll a[maxn * maxn][maxn * maxn]; 
int Det(int n) {
    ll ret = 1;
    int flg = 0;
    for(int i = 1; i <= n; ++i) {
        for(int j = i + 1; j <= n; ++j) {
            while(a[j][i]) {
                ll t = (a[i][i] % (a[j][i] * mod)) / a[j][i];
                for(int k = 1; k <= n; ++k) 
                    a[i][k] = (a[i][k] - a[j][k] * t % mod + mod) % mod;
                for(int k = 1; k <= n; ++k)
                    swap(a[i][k], a[j][k]);
                ++flg;
            }
        }
        if(a[i][i] == 0) return 0;
        ret = ret * a[i][i] % mod;
    }
    if(flg & 1) ret = -ret;
    ret %= mod;
    ret = (ret + mod) % mod;
    return ret;
}
int main()
{
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n, m, tot = 0;
    cin >> n >> m;
    for(int i = 1; i <= n; ++i) {
        for(int j = 1; j <= m; ++j) {
            cin >> g[i][j];
            if(g[i][j] == '.') 
                id[i][j] = ++tot;
        }
    }
    for(int i = 1; i <= n; ++i) {
        for(int j = 1; j <= m; ++j) {
            int d = 0;
            for(int k = 0; k < 4; ++k) {
                int x = i + dir[k][0], y = j + dir[k][1];
                if(x < 1 || x > n || y < 1 || y > m) continue;
                if(g[x][y] != '.') continue;
                d++;
                a[id[i][j]][id[x][y]] = -1;
            }
            a[id[i][j]][id[i][j]] = d;
        }
    }
    cout << Det(tot - 1) << endl;
}