Vietoris continuous selections and disconnectedness-like properties

Suppose that $X$ is a Hausdorff space such that its Vietoris hyperspace $({\mathcal{F}}(X),\tau _{V})$ has a continuous selection. Do disconnectedness-like properties of $X$ depend on the variety of continuous selections for $({\mathcal{F}}(X),\tau _{V})$ and vice versa? In general, the answer is ``yes'' and, in some particular situations, we were also able to set proper characterizations.