<p>You are given a positive integer <code>n</code> representing the number of nodes in an <strong>undirected</strong> graph. The nodes are labeled from <code>1</code> to <code>n</code>.</p>

<p>You are also given a 2D integer array <code>edges</code>, where <code>edges[i] = [a<sub>i, </sub>b<sub>i</sub>]</code> indicates that there is a <strong>bidirectional</strong> edge between nodes <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>. <strong>Notice</strong> that the given graph may be disconnected.</p>

<p>Divide the nodes of the graph into <code>m</code> groups (<strong>1-indexed</strong>) such that:</p>

<ul>
	<li>Each node in the graph belongs to exactly one group.</li>
	<li>For every pair of nodes in the graph that are connected by an edge <code>[a<sub>i, </sub>b<sub>i</sub>]</code>, if <code>a<sub>i</sub></code> belongs to the group with index <code>x</code>, and <code>b<sub>i</sub></code> belongs to the group with index <code>y</code>, then <code>|y - x| = 1</code>.</li>
</ul>

<p>Return <em>the maximum number of groups (i.e., maximum </em><code>m</code><em>) into which you can divide the nodes</em>. Return <code>-1</code> <em>if it is impossible to group the nodes with the given conditions</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/13/example1.png" style="width: 352px; height: 201px;" />
<pre>
<strong>Input:</strong> n = 6, edges = [[1,2],[1,4],[1,5],[2,6],[2,3],[4,6]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> As shown in the image we:
- Add node 5 to the first group.
- Add node 1 to the second group.
- Add nodes 2 and 4 to the third group.
- Add nodes 3 and 6 to the fourth group.
We can see that every edge is satisfied.
It can be shown that that if we create a fifth group and move any node from the third or fourth group to it, at least on of the edges will not be satisfied.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 3, edges = [[1,2],[2,3],[3,1]]
<strong>Output:</strong> -1
<strong>Explanation:</strong> If we add node 1 to the first group, node 2 to the second group, and node 3 to the third group to satisfy the first two edges, we can see that the third edge will not be satisfied.
It can be shown that no grouping is possible.
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 500</code></li>
	<li><code>1 &lt;= edges.length &lt;= 10<sup>4</sup></code></li>
	<li><code>edges[i].length == 2</code></li>
	<li><code>1 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt;= n</code></li>
	<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
	<li>There is at most one edge between any pair of vertices.</li>
</ul>

class Solution:
    def magnificentSets(self, n: int, edges: List[List[int]]) -> int:
        cn = []
        def ib(x, c):
            # print(x)
            nonlocal color, cn
            color[x] = c
            cn.append(x)
            for v in g[x]:
                if c == color[v]: return False
                if 0 == color[v]:
                    if not ib(v, -c): return False
            return True
        color = [0] * (n +4)
        g = defaultdict(list)
        for a,b in edges:
            g[a].append(b)
            g[b].append(a)
        ans = 0
        for start in range(1, n + 1):
            if color[start] != 0: continue
            cn.clear()
            if not ib(start, -1): return -1
            maxlength = 0
            for x in cn:
                visited = [0] * (n+1)
                q = deque()
                q.append((x, 1))
                while q:
                    u, length = q.popleft()
                    maxlength = max(maxlength, length)
                    if visited[u] == 0:
                        visited[u] = 1
                        for v in g[u]:
                            if visited[v] == 0:
                                q.append((v, length+1))
            ans+= maxlength
        return ans

20800