On the weighted safe set problem on paths and cycles

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.