  Given a continuum $X$, let $C(X)$ denote the hyperspace of all subcontinua of
$X$. In this paper we study the Vietoris hyperspace $NC^{*}(X)=\{ A \in
C(X):X\setminus A\text{ is connected}\}$ when $X$ is a finite graph or a
dendrite; in particular, we give conditions under which $NC^{*}(X)$ is compact,
connected, locally connected or totally disconnected. Also, we prove that if
$X$ is a dendrite and the set of endpoints of $X$ is dense, then $NC^{*}(X)$ is
homeomorphic to the Baire space of irrational numbers.
