GATE | GATE CS 2012 | Question 27

• Difficulty Level : Hard
• Last Updated : 28 Jun, 2021

Let G be a weighted graph with edge weights greater than one and G’be the graph constructed by squaring the weights of edges in G. Let T and T’ be the minimum spanning trees of G and G’, respectively, with total weights t and t’. Which of the following statements is TRUE?
(A) T’ = T with total weight t’ = t2
(B) T’ = T with total weight t’ < t2
(C) T’ != T but total weight t’ = t2
(D) None of the above

Explanation: Squaring the weights of the edges in a weighted graph will not change the minimum spanning tree. Assume the opposite to obtain a contradiction. If the minimum spanning tree changes then at least one edge from the old graph G in the old minimum spanning tree T must be replaced by a new edge in tree T’ from the graph G’ with squared edge weights. The new edge from G’ must have a lower weight than the edge from G. This implies that there exists some weights C1 and C2 such that C1 < C2 and C12 >= C22. This is a contradiction.
Source: http://www.cs.nyu.edu/courses/spring06/V22.0310-001/hw3.htm

Sums of squares of two or more numbers is always smaller than square of sum.
Example: 2^2 + 2^2 < (4)^2

But

there is one counter example when the graph has only one edge.
In that case, the two values are same.
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