Question

(a) Let T be a minimum spanning tree of a weighted graph G. Construct
a new graph G by adding a weight of k to every edge of G. Do the edges
of T form a minimum spanning tree of G. Prove the statement or give a
counterexample.

(b) Let P = {s, . . . , t} describe a shortest weighted path between
vertices s and t of a weighted graph G. Construct a new graph G by
adding a weight of k to every edge of G. Does P describe a shortest
path from s to t in G. Prove the statement or give a counterexample.

My solution:

a) Edges of T still form minimum spanning tree of G, since all edge weights are increased by same amount.

b) P still describes shortest path from s to t in G (same reason)

Can someone please verify the answers?

Answer 1

Although I don’t think SO is the best place for your question, your answer to question B is definitely wrong.

Consider a graph with 3 vertices (A,B,C), with the following edges:

A-B = 1
A-C = 0
C-B = 0

The shortest weighted path between A and B is A-C-B. If you add 2 to all the weights, your shortest path becomes A-B.

(Sorry, missed the first part of the question, there is an answer for that already by now. The reason why a is correct but b is wrong is that spanning trees always contain exactly n-1 edges, while the number of edges in a shortest weighted path may vary.)