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