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...

A distance-k dominating set S of a directed graph D is a set of vertices such that for every vertex v of D, there is a vertex u ∈ S at distance at most k from it. Minimum distance-k dominating set is especially important in communication networks for distributed data structure and for server placement. In this paper, we shall present simple distributed algorithms for finding the unique minimum ...

A distance-k dominating set S of a directed graph D is a set of vertices such that for every vertex v of D, there is a vertex u ∈ S at distance at most k from it. Minimum distance-k dominating set is especially important in communication networks for distributed data structure and for server placement. In this paper, we shall present simple distributed algorithms for finding the unique minimum ...

Let G = (V, E) be a graph without isolated vertices. A set D ⊆ V is a d-distance paired-dominating set of G if D is a d-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a d-distance paired-dominating set for graph G is the d-distance paired-domination number, denoted by γd p(G). In this paper, we study the ddistance paired-domination n...

We propose a fast, silent self-stabilizing protocol building a distance-k independent dominating set, named FID. The convergence of the protocol FID is established for any computation under the unfair distributed scheduler. The protocol FID reaches a terminal (also legitimate) configuration in at most 4n+k rounds, where n is the network size; it requires (k + 1)log(n + 1) bits per node. keyword...

We propose a memory efficient self-stabilizing protocol building k-independant dominating sets. A k-independant dominating set is a k-independant set and a k-dominating set. A set of nodes, I, is kindependent if the distance between any pair of nodes in I is at least k + 1. A set of nodes, D, is a k-dominating if every node is within distance k of a node of D. Our algorithm, named SID, is silen...

An exponential dominating set of graph $G = (V,E )$ is a subset $Ssubseteq V(G)$ such that $sum_{uin S}(1/2)^{overline{d}{(u,v)-1}}geq 1$ for every vertex $v$ in $V(G)-S$, where $overline{d}(u,v)$ is the distance between vertices $u in S$ and $v  in V(G)-S$ in the graph $G -(S-{u})$. The exponential domination number, $gamma_{e}(G)$, is the smallest cardinality of an exponential dominating set....

Suppose G = (V , E) is a simple graph and k is a fixed positive integer. A subset D ⊆ V is a distance k-dominating set of G if for every u ∈ V , there exists a vertex v ∈ D such that dG(u, v) ≤ k, where dG(u, v) is the distance between u and v in G. A set D ⊆ V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph G[D] contains a perfect matchin...