<p>Given a weighted undirected connected graph with <code>n</code>&nbsp;vertices numbered from <code>0</code> to <code>n - 1</code>,&nbsp;and an array <code>edges</code>&nbsp;where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>, weight<sub>i</sub>]</code> represents a bidirectional and weighted edge between nodes&nbsp;<code>a<sub>i</sub></code>&nbsp;and <code>b<sub>i</sub></code>. A minimum spanning tree (MST) is a subset of the graph&#39;s edges that connects all vertices without cycles&nbsp;and with the minimum possible total edge weight.</p>

<p>Find <em>all the critical and pseudo-critical edges in the given graph&#39;s minimum spanning tree (MST)</em>. An MST edge whose deletion from the graph would cause the MST weight to increase is called a&nbsp;<em>critical edge</em>. On&nbsp;the other hand, a pseudo-critical edge is that which can appear in some MSTs but not all.</p>

<p>Note that you can return the indices of the edges in any order.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/06/04/ex1.png" style="width: 259px; height: 262px;" /></p>

<pre>
<strong>Input:</strong> n = 5, edges = [[0,1,1],[1,2,1],[2,3,2],[0,3,2],[0,4,3],[3,4,3],[1,4,6]]
<strong>Output:</strong> [[0,1],[2,3,4,5]]
<strong>Explanation:</strong> The figure above describes the graph.
The following figure shows all the possible MSTs:
<img alt="" src="https://assets.leetcode.com/uploads/2020/06/04/msts.png" style="width: 540px; height: 553px;" />
Notice that the two edges 0 and 1 appear in all MSTs, therefore they are critical edges, so we return them in the first list of the output.
The edges 2, 3, 4, and 5 are only part of some MSTs, therefore they are considered pseudo-critical edges. We add them to the second list of the output.
</pre>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/06/04/ex2.png" style="width: 247px; height: 253px;" /></p>

<pre>
<strong>Input:</strong> n = 4, edges = [[0,1,1],[1,2,1],[2,3,1],[0,3,1]]
<strong>Output:</strong> [[],[0,1,2,3]]
<strong>Explanation:</strong> We can observe that since all 4 edges have equal weight, choosing any 3 edges from the given 4 will yield an MST. Therefore all 4 edges are pseudo-critical.
</pre>

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

<ul>
	<li><code>2 &lt;= n &lt;= 100</code></li>
	<li><code>1 &lt;= edges.length &lt;= min(200, n * (n - 1) / 2)</code></li>
	<li><code>edges[i].length == 3</code></li>
	<li><code>0 &lt;= a<sub>i</sub> &lt; b<sub>i</sub> &lt; n</code></li>
	<li><code>1 &lt;= weight<sub>i</sub>&nbsp;&lt;= 1000</code></li>
	<li>All pairs <code>(a<sub>i</sub>, b<sub>i</sub>)</code> are <strong>distinct</strong>.</li>
</ul>

class Solution:
    def findCriticalAndPseudoCriticalEdges(self, n: int, edges: List[List[int]]) -> List[List[int]]:
        vertices = list(range(n)) 

        # used as key to sort edges by weight
        def get_weight(edge):
            return edge[2]


        # union find functions
        def find(parent, v):
            if parent[v] != v:
                return find(parent, parent[v])
            else:
                return v
            
        def union(parent, v1, v2):
            parent[find(parent, vertices[v2])] = find(parent, vertices[v1])
            

        def sets_differ(parent, v1, v2):
            if find(parent, vertices[v1]) == find(parent, vertices[v2]):
                return False
            else:
                return True


        def kruskals_mod(edges, exclude, include): # exclude is an edge we are forcibly excluding, include is an edge we are forcibly adding.
            parent = list(range(n))
            weight = 0
            mst = []
            
            if include != None:
                source, destination = edges[include][0], edges[include][1]
                union(parent, source, destination)
                weight += edges[include][2]
                mst.append(edges[include])
                # adding this edge first makes it so when we fill the rest of the MST, it builds around the assumption that "include" is in the MST.

            for edge in edges:
                    if exclude != None:
                        if edge == edges[exclude]: # ignore the edge to exclude, don't add it to mst
                            continue
                    source, destination = edge[0], edge[1]
                    if sets_differ(parent, source, destination):
                        union(parent, source, destination)
                        weight += edge[2]
                        mst.append(edge)
                    else:
                        continue
    
            # check each edge to see if it's a cut edge
            for i in range(n):
                if sets_differ(parent, i, 0): # if the vertex i and 0 are in diff sets, then edge (i, 0) is a cut edge
                    return mst, float('inf') # returns weight of infinity if excluded edge is a cut edge
            return mst, weight
            

        for i in range(len(edges)): # adds edge number to each edge to keep track after sorting
            edges[i].append(i)

        edges.sort(key = get_weight)
        mst, mst_weight = kruskals_mod(edges, None, None)
        crit_edges = []
        pseudo_crit_edges = []

        for i in range(len(edges)):
            if kruskals_mod(edges, i, None)[1] > mst_weight: # if an edge's removal increases weight, the edge is critical
                crit_edges.append(edges[i][3])
            elif kruskals_mod(edges, None, i)[1] == mst_weight: # if an edge's forced inclusion is the same as mst weight, it's pseudocritical
                pseudo_crit_edges.append(edges[i][3])

        return [crit_edges, pseudo_crit_edges]
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