For any simple connected graph $G=(V,E)$, a defensive alliance is a subset $S$ of $V$ satisfying the condition that every vertex $vin S$ has at most one more neighbour in $V-S$ than it has in $S$. The minimum cardinality of any defensive alliance in $G$ is called the alliance number of $G$, denoted $a(G)$. In this paper, we introduce a new type of alliance number called $k$-strong alliance numb...

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 $G$ call...

Secure sets were introduced by Brigham, Dutton, and Hedetniemi [1] as an extension of defensive alliances in a graph. Members of a defensive alliance have agreed to come to the aid of their neighbors in the alliance for mutual protection from attack by nonmembers. Alliances can occur, for example, between nations, biological sequences, and business cartels, and also have been used to model a va...

Given the original set X where |X| = s, a sketch P is computed from X and made public. From another set Y where |Y | = s and P , we can reconstruct X if |X ∩ Y | ≥ |s − t|, where t < s is some threshold. The sketch P is secure if it does not reveal much information about X. A few constructions have been proposed, but they cannot handle multi-sets, that is, sets that may contain duplicate elemen...

Consider a robot that is navigating graph-based space and attempting to determine where it right now. To how distant from each group of fixed landmarks, can send signal. We discuss the problem determining minimum number landmarks necessary their optimal placement ensure always locate itself. The referred as graph's metric dimension, set nodes on which they are distributed known basis. On other ...