GATE CS 2016 Set 1 — Question 50
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Algorithms → Graph Theory (Math) → Kruskal's MST
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Question
is an undirected simple graph in which each edge has a distinct weight, and is a particular edge of . Which of the following statements about the minimum spanning trees (MSTs) of is/are TRUE?I. If is the lightest edge of some cycle in , then every MST of includes
II. If is the heaviest edge of some cycle in , then every MST of excludes
II. If is the heaviest edge of some cycle in , then every MST of excludes
Correct answer
(B) II only
Solution
Statement II is the standard Cycle Property of Minimum Spanning Trees: For any cycle in the graph, the edge with the maximum weight in that cycle is not in the MST (assuming distinct weights). Thus, Statement II is TRUE.Statement I is FALSE. The Cut Property states that if an edge is the minimum weight edge crossing some cut, it is in the MST. However, being the lightest edge in some cycle does not guarantee it is in the MST. Counter-example for I:
Consider a graph with vertices and two paths between them:
However, the MST will use path (weights 10, 20) to connect and because they are much smaller than 100. The edge forms a cycle with (edges ) where 100 is the heaviest edge, so is excluded from the MST.Thus, only Statement II is correct.
Consider a graph with vertices and two paths between them:
1.Path : a single edge with weight 100.
2.Path : edges with weight 10 and with weight 20.
3.Path : edges with weight 200 and with weight 300.
Consider the cycle formed by and : edges are . Here, is the lightest edge in cycle .However, the MST will use path (weights 10, 20) to connect and because they are much smaller than 100. The edge forms a cycle with (edges ) where 100 is the heaviest edge, so is excluded from the MST.Thus, only Statement II is correct.
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