On the weighted safe set problem on paths and cycles

Abstract

Let G be a graph, and let w be a positive real-valued weight function on V(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 $w(C) \ge w(D)$ 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$\mathrm{s}(G,w)$ and connected weighted safe number$\mathrm{cs}(G,w)$ 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), $\mathrm{s}(G,w) \le \mathrm{cs}(G,w)$ 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 32:954–974, 2011) concerning subgraph component polynomials for cycles and complete graphs.