<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:
        #process graph
        #if odd degree nodes > 4 return false
        #for each odd node, try connecting it to a previously unconnected odd node then remove those two from set
        #
        from collections import defaultdict
        dic = defaultdict(set)
        odd = set()
        for u, v in edges:
            dic[u].add(v)
            dic[v].add(u)
            if len(dic[u]) % 2:
                odd.add(u)
            else:
                odd.remove(u)
            if len(dic[v]) % 2:
                odd.add(v)
            else:
                odd.remove(v)
        odd = list(odd)
        if not len(odd):
            return True
        elif len(odd) == 2:
            #1 connect these two if unconnected 
            #check if there is any node i in 1, n s.t. both nodes are not connected to i
            u, v = odd[0], odd[1]
            if v not in dic[u]:
                return True
            for i in range(1, n+1):
                if i != u and i !=v:
                    if i not in dic[u] and i not in dic[v]:
                        return True
        elif len(odd) == 4:
            # create two pairs of unconnected nodes
            for i in range(4):
                for j in range(i+1, 4):
                    for k in range(4):
                        if k != j and k != i:
                            for l in range(4):
                                if l != j and l != i and l != k:
                                    if odd[j] not in dic[odd[i]] and odd[k] not in dic[odd[l]]:
                                        return True
        return False


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