What does $E(S)$ mean in the bipartite matching lecture?

Q: What does E(S) mean in the bipartite matching lecture?

E(S) denotes the neighborhood of a subset S of the left-side vertices in the right-side set. It is the set of all right-side vertices adjacent to at least one vertex in S.

E(S)

denotes the neighborhood of a subset S of the left-side vertices in the right-side set. It is the set of all right-side vertices adjacent to at least one vertex in S.

Conditions

S is a subset of the left-side vertices.
Adjacency is taken in the given bipartite graph.

Reasoning, step by step

Identify the subset S of the left-side vertices.
Find all right-side vertices that are connected by an edge to at least one vertex in S.
Collect these right-side vertices into the set $E(S)$ .

Example

The slide writes $E(S)$ on the right side and later states $E(S)$ ::= boys adjacent to at least one girl in S.

Common misconceptions

Confusing $E(S)$ with the set of edges incident to S.
Thinking that $E(S)$ includes vertices in S itself.

Watch the explanation

YouTube2.11.7 Bipartite Matching

2:36 – 3:16Watch this moment ↗

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