GATE CS 2026 Set 1 — Question 47
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Engineering Mathematics → Graph Theory (Math) → Graph Terminology (Degree, Paths, Cycles)
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Question
Let
G(V, E) be a simple, undirected graph. A vertex cover of is a subset such that for every , or . Let the size of the smallest vertex cover in be . Let be any vertex cover of size .For a vertex , which of the following constraints will always ensure that ?Correct answer
(A) The degree of v is at least k+1
Solution
We are given that is the size of the minimum vertex cover of . We want to find a condition on vertex that guarantees belongs to every minimum vertex cover .Consider Option (A): The degree of is at least .
Let be the set of neighbors of . We have .
Suppose, for the sake of contradiction, that there exists a minimum vertex cover (where ) such that .
Since is a vertex cover, for every edge incident to , the other endpoint must be in . That is, all neighbors of must be in .
Therefore, .
This implies .
But we are given that . This is a contradiction ( is false).
Thus, our assumption that must be false. So, must be in .Option (A) is correct.
Let be the set of neighbors of . We have .
Suppose, for the sake of contradiction, that there exists a minimum vertex cover (where ) such that .
Since is a vertex cover, for every edge incident to , the other endpoint must be in . That is, all neighbors of must be in .
Therefore, .
This implies .
But we are given that . This is a contradiction ( is false).
Thus, our assumption that must be false. So, must be in .Option (A) is correct.
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