GATE CS 2018 Set 1 — Question 57
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Algorithms → Graph Theory (Math) → Kruskal's MST
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Question
Consider the following undirected graph G:
Choose a value for that will maximize the number of minimum weight spanning trees (MWSTs) of G. The number of MWSTs of G for this value of is ______.
Correct answer
4 to 4
Solution
To find the number of Minimum Weight Spanning Trees (MWSTs), we use Kruskal's algorithm. The edges in increasing order of weight are:
1.Weight 1: 1 edge
2.Weight 3: 2 edges
3.Weight 4: 4 edges (plus potentially if )
4.Weight 5: 1 edge
Let's analyze the cycles:- Cycle 1: (Top, Mid-Left, Bottom-Center) with weights 4, 3, 1. To avoid a cycle, we must exclude one edge. The MWST will always include the edges of weight 1 and 3, and potentially the edge of weight 4 if needed for connectivity.
- Cycle 2: (Top, Mid-Right, Bottom-Center) with weights , 3, 1. Similarly, MWST will include 1 and 3.
- Cycle 3: (Mid-Left, Bottom-Left, Bottom-Center) with weights 4, 4, 3. To connect Bottom-Left, we must pick one of the weight 4 edges. There are 2 choices.
- Cycle 4: (Mid-Right, Bottom-Right, Bottom-Center) with weights 5, 4, 3. To connect Bottom-Right, we must pick the weight 4 edge (Bottom-Center to Bottom-Right) as it's smaller than 5.
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