Can a possible subgraph contain a completely unconnected vertex?

Question

If you have say, a square. Can a subgraph of that square be the top left vertex left alone with an order of 0, whilst the right-most and the bottom lines are included. Would that be valid? Or would it have to be only the left and right lines.

Answer 1

Yes, a subgraph can contain an isolated vertex. You can have any subset of the vertices, and any subset of the edges, provided only that any vertices incident to the edges be in the subgraph.

Answer 2

A subgraph of a graph $(V,E)$ is a graph $(V',E')$ where $V'\subseteq V$ and $E'\subseteq E$. For example it is even allowed to have $V'=V$ and $E'=\emptyset$. (But of course we can only have $vw\in E'$ if both $v\in V'$ and $w\in V'$).

What you are perhaps thinking of is an induced subgraph, that is a subgraph where $E'=\{\,vw\in E\mid v\in V', w\in V'\,\}$.