What does the Bottleneck Lemma say?

Conditions

• There is a subset S of the left-side vertices.
• |S| > |$E(S)$|.

Reasoning, step by step

1. Identify if there is a bottleneck S in the graph.
2. If such a subset S exists where |S| > |$E(S)$|, conclude that no matching is possible.

Example

The slide title reads Bottleneck Lemma and the statement reads If there is a bottleneck, then no match is possible obviously.

Common misconceptions

• Thinking that the lemma provides a method to find the matching.
• Assuming that the absence of a bottleneck automatically means a matching exists without further proof.

Watch the explanation

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• What is Hall's theorem as introduced at the end of the clip?
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