  Let $G$ be a graph, and let $w: V(G) \to \mathbb{R}$ be a weight function on
the vertices of $G$. For every subset $X$ of $V(G)$, let $w(X)=\sum_{v \in X}
w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$
if, for every component $C$ of the subgraph induced by $S$ and every component
$D$ of $G-S$, we have $w(C) \geq w(D)$ whenever there is an edge between $C$
and $D$. If the subgraph of $G$ induced by a weighted safe set $S$ is
connected, then the set $S$ is called a connected weighted safe set of $(G,w)$.
The weighted safe number $s(G,w)$ and connected weighted safe number $cs(G,w)$
of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all
connected weighted safe sets of $(G,w)$, respectively. It is easy to see that
for any pair $(G,w)$, ${s}(G,w) \le {cs}(G,w)$ by their definitions. In this
paper, we discuss the possible equality when $G$ is a path or a cycle. We also
give an answer to a problem due to Tittmann et al. [Eur. J. Combin. Vol. 32
(2011)] concerning subgraph component polynomials for cycles and complete
graphs.
