What is a minimum vertex separator as in this definition? In a research paper the following definition appears that I'm not able to understand completely.

Let $G=(V,E)$ be an undirected unweighted graph with vertex set $V$ and edge set $E$, no self-loops, no parallel-edges.
Let $\{ u, v\} \in \binom{V}{2}$ be a vertex pair over the set of all vertices pairs. Let $\textrm{Sep}(u, v)$ be a minimum $u-v$ vertex separator in $G$ if $\{u, v\} \notin E$ or in $G' = (V,E \setminus \{u,v\}) $ otherwise.

It is difficult for me to figure out what this definition imply.

In particular can $\textrm{Sep}(u,v)$ be an empty set?
It seems that $Sep(u, v)$ is a vertex separator extended to graphs where $u$ and $v$ are adjacent, by ignoring their common edge.

I assume the correct definition is  


  ... vertex separator in $G$ if $\{u,v\} \notin E$ or in $G′=(V,E \setminus \{\{u,v\}\})$ otherwise.


As given, the definition makes no sense. There is no point in removing $\{u, v\}$ from $E$ if $\{u, v\} \notin E$ and, of course, $\{u, v\} \not\subset E$.

In this case, $Sep(u, v)$ can be $\emptyset$ if $\{u, v\}$ is the only edge connecting two components, or if the graph had two components to begin with.