Given an undirected tree T = (V, E) and an integer k. Give a polynomial time algorithm which outputs ”yes” if T has a vertex cover of size at most k and ”no” otherwise. Prove that your algorithm is correct and that it runs in polynomial time.

Write a proof that for that if a tree has n vertices then it must have n-1 edges.

Hey, Kruskal's algorithm can return different spanning trees for the  input Graph G.Show that for every minimal spanning tree T of G, there is an execution of the algorithm that gives T as a result.   How can i do that? Thank you in advance!

You are given a weighted, undirected graph G = (V, E) which is guaranteed to be connected. Design an algorithm which runs in O(V E + V 2 log V ) time and determines which of the edges appear in all minimum spanning trees of G. Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain English.Please give total time complexity. test whether a particular edge appears in all MSTs