SAFE ETERNAL 1-SECURE SETS IN GRAPHS

An eternal $1$-secure set, in a graph $G = (V, E)$ is set $D \subset V$ having the property that for any finite sequence of vertices $r_1, r_2, \ldots, r_k$ there exists $v_1, v_2, v_k$ and $ D D_0, D_1, D_2, D_k$ dominating sets $G$, such each $i$, $1 \leq i k$, $D_{i} (D_{i-1} - \{v_i\}) \cup \{r_i\}$, where $v_i \in D_{i-1}$ $r_i N[v_i]$. Here v_i$ possible. The cardinality smallest $G$ called $1$-security number $G$. In this paper we study variations named safe sets. A vertex $v$ with respect to an $S$ if $N[v] \bigcap S =1$. 1 secure at least one $S$. We characterize class graphs which all excluding those are safe. Also introduce new kind directed represent transformation from another 1-secure given its properties.