<p>You are given an <code>m x n</code> binary grid <code>grid</code> where <code>1</code> represents land and <code>0</code> represents water. An <strong>island</strong> is a maximal <strong>4-directionally</strong> (horizontal or vertical) connected group of <code>1</code>&#39;s.</p>

<p>The grid is said to be <strong>connected</strong> if we have <strong>exactly one island</strong>, otherwise is said <strong>disconnected</strong>.</p>

<p>In one day, we are allowed to change <strong>any </strong>single land cell <code>(1)</code> into a water cell <code>(0)</code>.</p>

<p>Return <em>the minimum number of days to disconnect the grid</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/12/24/land1.jpg" style="width: 500px; height: 169px;" />
<pre>
<strong>Input:</strong> grid = [[0,1,1,0],[0,1,1,0],[0,0,0,0]]

<strong>Output:</strong> 2
<strong>Explanation:</strong> We need at least 2 days to get a disconnected grid.
Change land grid[1][1] and grid[0][2] to water and get 2 disconnected island.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/12/24/land2.jpg" style="width: 404px; height: 85px;" />
<pre>
<strong>Input:</strong> grid = [[1,1]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> Grid of full water is also disconnected ([[1,1]] -&gt; [[0,0]]), 0 islands.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 30</code></li>
	<li><code>grid[i][j]</code> is either <code>0</code> or <code>1</code>.</li>
</ul>

class Solution {
public:
    int dr[4] = {1,-1,0,0};
    int dc[4] = {0,0,1,-1};
    int R, C;
    int num_island = 1; // 1st island start with idx = 2

    // articulation points
    int n = 1; // 1-n
    int idx = 1;
    vector<vector<int>> adj;
    vector<bool> visited;
    vector<int> disc;
    vector<int> low;
    vector<bool> ans;

    void ap(int u, int par) {
        visited[u] = true;
        disc[u] = low[u] = idx++;
        int child = 0;

        for(auto& nu : adj[u]) {
            if(!visited[nu]) {
                child++;
                ap(nu, u);
                low[u] = min(low[u], low[nu]);
                if(par != -1 && disc[u] <= low[nu]) ans[u] = true;
            } else if(nu != par) {
                low[u] = min(low[u], disc[nu]);
            }
        }

        if(par == -1 && child > 1) ans[u] = true;
    }

    int solve(vector<vector<int>>& grid) {
        // numbering
        for(int r = 0; r < R; r++) {
            for(int c = 0; c < C; c++) {
                if(grid[r][c] != 0) grid[r][c] = n++;
            }
        }

        // only 1 data
        if(n-1 == 1) return 1; 

        // pre
        adj.assign(n, vector<int>());
        visited.assign(n, false);
        disc.assign(n, 0);
        low.assign(n, 0);
        ans.assign(n, false);
        
        // adj
        for(int r = 0; r < R; r++) {
            for(int c = 0; c < C; c++) {
                if(grid[r][c] == 0) continue;
                for(int i = 0; i < 4; i++) {
                    int nr = r + dr[i];
                    int nc = c + dc[i];
                    if(nr < 0 || nc < 0 || nr == R || nc == C) continue;
                    if(grid[nr][nc] == 0) continue;
                    int a = grid[r][c];
                    int b = grid[nr][nc];
                    adj[a].push_back(b);
                    adj[b].push_back(a);
                }
            }
        }

        for(int i = 1; i < n; i++) {
            if(!visited[i]) ap(i, -1);
        }

        int total_ap = 0;
        for(int i = 1; i < n; i++) {
            if(ans[i]) total_ap++;
        }

        if(total_ap >= 1) return 1;
        return 2;
    }

    void dfs(int r, int c, vector<vector<int>>& grid) {
        grid[r][c] = num_island;
        for(int i = 0; i < 4; i++) {
            int nr = r + dr[i];
            int nc = c + dc[i];
            if(nr < 0 || nc < 0 || nr == R || nc == C) continue;
            if(grid[nr][nc] != 1) continue;
            dfs(nr, nc, grid);
        }
    }

    int minDays(vector<vector<int>>& grid) {
        R = grid.size(), C = grid[0].size();

        // check how many island via idx
        for(int r = 0; r < R; r++) {
            for(int c = 0; c < C; c++) {
                if(grid[r][c] == 1) {
                    num_island++;
                    dfs(r, c, grid);
                }
            }
        }

        if(num_island - 1 != 1) return 0;
        return solve(grid);
    }
};
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