A mobile ad-hoc network (MANET), a set of wirelessly connected sensor nodes, is a dynamic system that executes hop-by-hop routing independently with no external help of any infrastructure. Proper selection of cluster heads can increase the life time of the Ad-hoc network by decreasing the energy consumption. Although different methods have been successfully proposed by researchers to tackle...

Let $G$ be a connected graph. A set $S$ of vertices in is 1-movable 2-resolving hop dominating if and for every $v \in S$, either $S\backslash \{v\}$ or there exists vertex $u \big((V (G) \backslash S) \cap N_G(v)\big)$ such that $\big(S \{v\}\big) \cup \{u\}$ $G$. The domination number $G$, denoted by $\gamma^{1}_{m2Rh}(G)$ the smallest cardinality In this paper, we investigate concept study i...

Let $G=(V,E)$ be a simple graph. A set $Dsubseteq V$ is adominating set of $G$ if every vertex in $Vsetminus D$ has atleast one neighbor in $D$. The distance $d_G(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$G$. An $(u,v)$-path of length $d_G(u,v)$ is called an$(u,v)$-geodesic. A set $Xsubseteq V$ is convex in $G$ ifvertices from all $(a, b)$-geodesics belon...