Alice and Bob have an undirected graph of n nodes and three types of edges:Type 1: Can be traversed by Alice only.Type 2: Can be traversed by Bob only.Type 3: Can be traversed by both Alice and Bob.Given an array edges where edges[i] = [typei, ui, vi] represents a bidirectional edge of type typei between nodes ui and vi, find the maximum number of edges you can remove so that after removing the edges, the graph can still be fully traversed by both Alice and Bob. The graph is fully traversed by Alice and Bob if starting from any node, they can reach all other nodes.Return the maximum number of edges you can remove, or return -1 if Alice and Bob cannot fully traverse the graph. Example 1:Input: n = 4, edges = [[3,1,2],[3,2,3],[1,1,3],[1,2,4],[1,1,2],[2,3,4]]Output: 2Explanation: If we remove the 2 edges [1,1,2] and [1,1,3]. The graph will still be fully traversable by Alice and Bob. Removing any additional edge will not make it so. So the maximum number of edges we can remove is 2.Example 2:Input: n = 4, edges = [[3,1,2],[3,2,3],[1,1,4],[2,1,4]]Output: 0Explanation: Notice that removing any edge will not make the graph fully traversable by Alice and Bob.Example 3:Input: n = 4, edges = [[3,2,3],[1,1,2],[2,3,4]]Output: -1Explanation: In the current graph, Alice cannot reach node 4 from the other nodes. Likewise, Bob cannot reach 1. Therefore it's impossible to make the graph fully traversable.  Constraints:1 <= n <= 1051 <= edges.length <= min(105, 3 * n * (n - 1) / 2)edges[i].length == 31 <= typei <= 31 <= ui < vi <= nAll tuples (typei, ui, vi) are distinct.

You have an undirected, connected graph of n nodes labeled from 0 to n - 1. You are given an array graph where graph[i] is a list of all the nodes connected with node i by an edge.Return the length of the shortest path that visits every node. You may start and stop at any node, you may revisit nodes multiple times, and you may reuse edges. Example 1:Input: graph = [[1,2,3],[0],[0],[0]]Output: 4Explanation: One possible path is [1,0,2,0,3]Example 2:Input: graph = [[1],[0,2,4],[1,3,4],[2],[1,2]]Output: 4Explanation: One possible path is [0,1,4,2,3] Constraints:n == graph.length1 <= n <= 120 <= graph[i].length < ngraph[i] does not contain i.If graph[a] contains b, then graph[b] contains a.The input graph is always connected.

The edge connectivity of an undirected graph is the minimum number k of edgesthat must be removed to disconnect the graph. For example, the edge connectivityof a tree is 1, and the edge connectivity of a cyclic chain of vertices is 2. Showhow to determine the edge connectivity of an undirected graph G D .V; E/ byrunning a maximum-flow algorithm on at most jV j flow networks, each havingO.V / vertices and O.E/ edges.

Problem StatementYou are roaming in a forest. This forest can be viewed as a graph with n nodes and m undirected edges. You roam the forest by traveling through the undirected edges from one node to another. You want to travel all the nodes at least once and stop when you have visited every node. Given that this is a connected graph, find the smallest lexicographical sequence of nodes that can be visited and return it in an array.The m undirected edges are given in the form of two arrays a and b, wherein there is an undirected edge between a[i] and b[i].Input FormatThe first line contains n, the number of nodes in the forest.The second line contains m, the number of edges in the forest.The next m lines contain the elements of the array a.The next line contains m, the number of edges in the forest.The next m lines contain the elements of the array b.Constraints2 <= n <= 1051 <= m <= 105The graph is a single connected component.If i!=j then (a[i], b[i]) != (a[j], b[j])Output FormatReturn an array denoting the lexicographically smallest sequence in which the nodes can be visited.Evaluation ParametersSample Input3211223Sample Output123

There is an undirected graph of n nodes. You are given a 2D array edges, where edges[i] = [ui, vi, lengthi] describes an edge between node ui and node vi with a traversal time of lengthi units.Additionally, you are given an array disappear, where disappear[i] denotes the time when the node i disappears from the graph and you won't be able to visit it.Notice that the graph might be disconnected and might contain multiple edges.Return the array answer, with answer[i] denoting the minimum units of time required to reach node i from node 0. If node i is unreachable from node 0 then answer[i] is -1. Example 1:Input: n = 3, edges = [[0,1,2],[1,2,1],[0,2,4]], disappear = [1,1,5]Output: [0,-1,4]Explanation:We are starting our journey from node 0, and our goal is to find the minimum time required to reach each node before it disappears.For node 0, we don't need any time as it is our starting point.For node 1, we need at least 2 units of time to traverse edges[0]. Unfortunately, it disappears at that moment, so we won't be able to visit it.For node 2, we need at least 4 units of time to traverse edges[2].

Consider an undirected simple graph G with 300 nodes. The maximum number of edges to be included in G so that the graph is not connected is?44511 14800 4401134511