GATE CS 2021 Set 2 — Question 11
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Algorithms → Graph Theory (Math) → Kruskal's MST
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Question
Let be a connected undirected weighted graph. Consider the following two statements.
: There exists a minimum weight edge in which is present in every minimum spanning tree of .
: If every edge in has distinct weight, then has a unique minimum spanning tree.Which one of the following options is correct?
: There exists a minimum weight edge in which is present in every minimum spanning tree of .
: If every edge in has distinct weight, then has a unique minimum spanning tree.Which one of the following options is correct?
Correct answer
(C) S₁ is false and S₂ is true.
Solution
is false. Consider a cycle where all edges have the same minimum weight. Any minimum spanning tree (MST) will exclude one of these edges to avoid a cycle. Therefore, no single minimum weight edge is guaranteed to be present in every MST. is true. It is a standard theorem in graph theory that if all edge weights in a connected graph are distinct, then the graph has a unique minimum spanning tree.
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