GATE CS 2021 Set 1 — Question 27
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Algorithms → Graph Theory (Math) → Kruskal's MST
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Question
Consider the following undirected graph with edge weights as shown:
The number of minimum-weight spanning trees of the graph is _______.
Correct answer
3 to 3
Solution
Label the 3x3 grid of vertices by row (top, middle, bottom) and column (left, centre, right). From the figure:
- Top row horizontals: 0.1, 0.1. Middle row horizontals: 0.1 (left), 0.9 (right). Bottom row horizontals: 0.9 (left), 0.1 (right).
- Upper verticals (top to middle): 0.9, 0.9, 0.9. Lower verticals (middle to bottom): 0.1, 0.1, 0.1.
- the top row (3 vertices, joined by its two 0.1 edges), which is a tree;
- the middle and bottom rows (6 vertices), joined by middle-left (0.1), bottom-right (0.1) and the three lower verticals (0.1). That is 5 edges on 6 vertices with no cycle, so it is also a tree.
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