Practice Case 2: K-sized Subset With Exactly K Neighbours (26.1.2.2.2)

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8 questions

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6 sample questions

Question 1.
What is a bipartite graph?
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Think about how the vertices can be split.
Question 2.
Define Hall's Marriage Theorem in simple terms.
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Focus on the terms 'matching' and 'condition'.
3Question 3.
What is required for a complete matching to exist according to Hall's theorem?
- |N(A)| < |A|
- |N(A)| = |A|
- |N(A)| ≥ |A|
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Remember the conditions that must hold.
4Question 4.
True or False: A bipartite graph cannot have a complete matching if any subset has fewer neighbours than its size.
- True
- False
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Think about Hall's necessary condition.
Question 5.
Construct a bipartite graph with 5 nodes in V1 and 6 in V2, and illustrate a case where Hall's condition does not hold true.
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Make sure to check subsets carefully.
Question 6.
Prove that a complete matching exists in a bipartite graph where every student is guaranteed to have at least 2 job offers.
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Think about matching strategies.

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