On the Connected Safe Number of Some Classes of Graphs

Abstract

For a connected simple graph $\mathcal{G}$ , a nonempty subset $\mathcal{S}$ of $V\left(\mathcal{G}\right)$ is a connected safe set if the induced subgraph $\mathcal{G}\left[\mathcal{S}\right]$ is connected and the inequality $\left|\mathcal{S}\right|\ge \left|\mathcal{D}\right|$ satisfies for each connected component $\mathcal{D}$ of $\mathcal{G\setminus S}$ whenever an edge of $\mathcal{G}$ exists between $\mathcal{S}$ and $\mathcal{D}$ . A connected safe set of a connected graph $\mathcal{G}$ with minimum cardinality is called the minimum connected safe set and that minimum cardinality is called the connected safe numbers. We study connected safe sets with minimal cardinality of the ladder, sunlet, and wheel graphs.