Graph neighbourhood ($N(S)$)

The neighbourhood $N(S)$ of a vertex set $S$ consists of all vertices adjacent to at least one vertex of $S$.

A neighbour of a vertex $v$ in a graph is a vertex joined to $v$ by an edge. These vertices form the graph neighbourhood $N(v)$.

The closed neighbourhood of a vertex set is $N[S]=S\cup N(S)$.

The external vertex boundary of a vertex set $S$ isFor comparisons at fixed $|S|$, minimizing $|{\partial }_{v}S|$ is equivalent to minimizing $|N[S]|$.

A common neighbour of vertices $u$ and $v$ is adjacent to both of them.