A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...

a set $s$ of vertices in a graph $g$ is a dominating set if every vertex of $v-s$ is adjacent to some vertex in $s$. the domination number $gamma(g)$ is the minimum cardinality of a dominating set in $g$. the annihilation number $a(g)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $g$ is at most the number of edges in $g$. in this p...

‎‎Let $G=(V‎, ‎E)$ be a simple graph with vertex set $V$ and edge set $E$‎. ‎A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacent‎‎or incident to at least one element $yin Vcup E$ for which $f(y)=2$‎. ‎The weight of an‎‎MRDF $f$ is $sum _{xin Vcup E} f(x)$‎. ‎The mi...

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

We propose a memory efficient self-stabilizing protocol building k-independent dominating sets. A k-independent dominating set is a k-independent set and a kdominating set. A set of nodes, I, is k-independent 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...

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

We present a fixed parameter algorithm that constructively solves the k-dominating set problem on graphs excluding one of the K5 or K3,3 as a minor in time O(3 6 √ n). In fact, we present our algorithm for any H-minor-free graph where H is a single-crossing graph (can be drawn on the plane with at most one crossing) and obtain the algorithm for K3,3(K5)-minor-free graphs as a special case. As a...

This paper studies the computational complexity of the Edge Packing problem and the Vertex Packing problem. The edge packing problem (denoted by EDS ) and the vertex packing problem (denoted by DS ) are linear programming duals of the edge dominating set problem and the dominating set problem respectively. It is shown that these two problems are equivalent to the set packing problem with respec...

We propose a memory efficient self-stabilizing protocol building k-independent dominating sets. A k-independent dominating set is a k-independent set and a kdominating set. A set of nodes, I, is k-independent 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...

We propose a memory efficient self-stabilizing protocol building k-independent dominating sets. A k-independent dominating set is a k-independent set and a kdominating set. A set of nodes, I, is k-independent 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...