<p>There is an <strong>undirected</strong> graph consisting of <code>n</code> nodes numbered from <code>1</code> to <code>n</code>. You are given the integer <code>n</code> and a <strong>2D</strong> array <code>edges</code> where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> indicates that there is an edge between nodes <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>. The graph can be disconnected.</p>

<p>You can add <strong>at most</strong> two additional edges (possibly none) to this graph so that there are no repeated edges and no self-loops.</p>

<p>Return <code>true</code><em> if it is possible to make the degree of each node in the graph even, otherwise return </em><code>false</code><em>.</em></p>

<p>The degree of a node is the number of edges connected to it.</p>

<p>&nbsp;</p>
<p><strong>Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/26/agraphdrawio.png" style="width: 500px; height: 190px;" />
<pre>
<strong>Input:</strong> n = 5, edges = [[1,2],[2,3],[3,4],[4,2],[1,4],[2,5]]
<strong>Output:</strong> true
<strong>Explanation:</strong> The above diagram shows a valid way of adding an edge.
Every node in the resulting graph is connected to an even number of edges.
</pre>

<p><strong>Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/26/aagraphdrawio.png" style="width: 400px; height: 120px;" />
<pre>
<strong>Input:</strong> n = 4, edges = [[1,2],[3,4]]
<strong>Output:</strong> true
<strong>Explanation:</strong> The above diagram shows a valid way of adding two edges.</pre>

<p><strong>Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/26/aaagraphdrawio.png" style="width: 150px; height: 158px;" />
<pre>
<strong>Input:</strong> n = 4, edges = [[1,2],[1,3],[1,4]]
<strong>Output:</strong> false
<strong>Explanation:</strong> It is not possible to obtain a valid graph with adding at most 2 edges.</pre>

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

<ul>
	<li><code>3 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>2 &lt;= edges.length &lt;= 10<sup>5</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 are no repeated edges.</li>
</ul>

class Solution:
    def isPossible(self, n: int, edges: List[List[int]]) -> bool:
        # each degree
        # 1: 2, 4
        # 2: 1, 3, 4, 5
        # 3: 2, 4
        # 4: 1, 2, 3, (5)
        # 5: 2, (4)
        graph = {i:set() for i in range(1, n+1)}
        # find all odd
        for i,j in edges:
            graph[i].add(j)
            graph[j].add(i)
        # odd+odd
        odds = {k for k in graph if len(graph[k]) % 2 == 1}
        # odd+even+odd
        if len(odds) == 0:
            return True
        # 0 odd: true
        
        # 1 odd: not possible
        if len(odds) % 2 == 1:
            return False
        # 2 odd: odd1+odd2 or odd1+even+odd
        if len(odds) == 2:
            odd1, odd2 = tuple(odds) 
            # 2.1: odd1 and odd2 not connected => true: O(1)
            if odd1 not in graph[odd2] and odd2 not in graph[odd1]:
                return True
            # 2.2: exists and even that not connect to odd1 and odd2: O(N)
            else:
                for k in graph:
                    if k in odds:
                        continue
                    if odd1 not in graph[k] and odd2 not in graph[k]:
                        return True
                return False
        # 3 odd: not possible
        # 4 odd: odd1+odd2; odd3+odd4; O(1)
        if len(odds) == 4:
            for odd1, odd2, odd3, odd4 in permutations(tuple(odds)):
                if (odd1 not in graph[odd2] and odd2 not in graph[odd1]) and (odd3 not in graph[odd4] and odd4 not in graph[odd3]):
                    return True
            return False
        return False
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