The subgraph induced by the neighbourhood of a vertex

Question

I have the connected graph as in picture 1 attached.

Is it correct if I take the subgraph induced by the neighbourhood of the vertex "a" as in the picture 2 below?

The neighbouring vertices of "a" are the vertices circled in red, right?

And the induced subgraph in Picture 2 is a connected graph right?

Can someone please help me with this question.

Thanks a lot in advance.

Answer 1

Normally the subgraph induced by a set of vertices means those vertices and all edges between them that were in the original graph.

So in your case it would be the four red vertices and the one edge between the topmost two of them.

There is also sometimes a distinction between "open neibourhood" (all vertices adjacent to $a$) and "closed neibourhood" (all those and also $a$).