GATE CS 2016 Set 1 — Question 49
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Algorithms → Graph Theory (Math) → Kruskal's MST
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Question
Let be a complete undirected graph on 4 vertices, having 6 edges with weights being 1, 2, 3, 4, 5, and 6. The maximum possible weight that a minimum weight spanning tree of can have is _________.
Correct answer
7 to 7
Solution
We want to assign the weights to the edges of such that the Minimum Spanning Tree (MST) weight is maximized.Kruskal's algorithm adds edges in increasing order of weight unless they form a cycle.
We want to force the algorithm to reject smaller weights (3, 4, ...) so that a larger weight is chosen.
1.Edge with weight 1 is always added (no cycle possible with 1 edge).
2.Edge with weight 2 is always added (no cycle possible with 2 edges in a simple graph).
Current MST weight = . The MST for 4 vertices must have 3 edges. We need to select one more edge.We want to force the algorithm to reject smaller weights (3, 4, ...) so that a larger weight is chosen.
- Can we reject 3? Yes, if edges 1, 2, and 3 form a triangle (cycle of length 3). Let vertices be . Edges . Edge 3 forms a cycle and is rejected.
- Now we have a connected component and an isolated vertex . We need to connect to the component.
- The available edges connecting to are .
- The remaining weights are . These must be assigned to these three edges.
- Kruskal's algorithm will pick the smallest available edge to connect the components. The smallest remaining weight is 4.
- Edge 4 does not form a cycle (it connects a new vertex). So 4 is added.
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