Indicate a set of edges of G that form a spanning tree. List the edges.
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- vector<VertexT> Vertices; map<VertexT, vector<list>> adjList; What is the worst case time complexity for this version of _LookupVertex? bool _LookupVertex(VertexT v) const { if (adjList.count(v) != 0) { return true; } return false; } V is the number of total vertices in the graph, E is the average number of edges per vertex. O(v) O(VlogE) O(logV) O(V^2) O(E+V)Is it possible for a minimum spanning tree to have a cycle? Why or why not?True or false help
- Please provide all the spanning trees possible from the graph below:Kruskal's Algorithm: Kruskal's algorithm is a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest. This algorithm is a Greedy Algorithm. The steps to find a MST using this algorithm are as follows: Sort all the edges in non-decreasing order of their weight. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it. Repeat step2 until there are (V-1) edges in the spanning tree. Example with 10 vertices and 20 weighted edges:A complete binary tree is a graph defined through the following recursive definition. Basis step: A single vertex is a complete binary tree. Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the the graph formed by starting with a root r, and adding an edge from r to each of the vertices r1, r2 is also a complete binary tree. The set of leaves of a complete binary tree can also be defined recursively. Basis step: The root r is a leaf of the complete binary tree with exactly one vertex r. Inductive step: The set of leaves of the tree T built from trees T1, T2 is the union of the sets of leaves of T1 and the set of leaves of T2. The height h(T) of a binary tree is defined in the class. Use structural induction to show that `(T), the number of leaves of a complete binary tree T, satisfies the following inequality `(T) ≤ 2 h(T) .