GATE CS 2014 Set 2 — Question 62
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Algorithms → Graph Theory (Math) → Kruskal's MST
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Question
The number of distinct minimum spanning trees for the weighted graph below is ______
Correct answer
6 to 6
Solution
To find the number of distinct Minimum Spanning Trees (MSTs), we analyze the edges by weight:
1.Edges with weight 1: There are 3 edges with weight 1. These edges do not form a cycle among themselves. According to the properties of MSTs, if the edges of a certain weight do not form a cycle, they must all be included in every MST (assuming the graph is connected and we are building the tree from minimum weights up). So, all 3 edges of weight 1 are included.
2.Edges with weight 2: The graph has 5 vertices (Top, Middle-Left, Middle-Right, Bottom-Left, Bottom-Right). An MST must have edges. Since we have already selected 3 edges of weight 1, we need exactly more edge. This edge must be of weight 2.
3.Selection: We must choose 1 edge from the available edges of weight 2 such that it does not form a cycle with the already selected weight 1 edges. By inspecting the graph (or using the Matrix Tree Theorem on the contracted graph), it turns out there are exactly 6 valid choices for this final edge that connect the components without creating a cycle.
Thus, there are 6 distinct Minimum Spanning Trees.Continue learning with Success Tracker
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