GATE CS 2018 Set 1 — Question 53
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Engineering Mathematics → Graph Theory (Math) → Connectivity
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Question
Let be a graph with vertices, with each vertex labelled by a distinct permutation of the numbers . There is an edge between vertices and if and only if the label of can be obtained by swapping two adjacent numbers in the label of . Let denote the degree of a vertex in , and denote the number of connected components in . Then,
Correct answer
109 to 109
Solution
1.Degree : A vertex represents a permutation of length . An edge exists between two vertices if one can be obtained from the other by swapping two adjacent elements. In a sequence of length 100, there are pairs of adjacent positions. Swapping any of these 99 pairs results in a unique neighbor. Therefore, the degree of every vertex is .
2.Connected components : It is a fundamental property of permutations that any permutation can be transformed into any other permutation using a sequence of adjacent swaps (e.g., via the bubble sort algorithm). Since every vertex can reach every other vertex, the graph is connected. Thus, the number of connected components is .
3.Final Calculation: .
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