Let G be a graph. A set S ⊆ V (G) is hop dominating of if for every v ∈ (G)\S, there exists u such that dG(u, v) = 2. The minimum cardinality γh(G) the domination number G. Any γh-set which intersects transversal set. γbh(G) in In this paper, we initiate study domination. First, characterize graphs whose values are either n or − 1, and determine specific some graphs. Next, show positive integer...


 Let G be a connected graph. A set S of vertices in is 2-resolving hop dominating if and for every vertex x ∈ V (G)\S there exists y such that dG(x, y) = 2. The minimum cardinality called the domination number denoted by γ2Rh(G). This study aims to combine concept with sets graphs. main results generated this include characterization join, corona lexicographic product two graphs, as well ...

Let G be a connected graph. A set W ⊆ V (G) is resolving hop dominating of if in and for every vertex v ∈ \ there exists u such that dG(u, v) = 2. S 1-movable S, either {v} or ((V S) ∩ NG(v)) (S {v}) ∪ {u} G. The domination number G, denoted by γ 1 mRh(G) the smallest cardinality This paper presents characterization sets join, corona lexicographic product graphs. Furthermore, this determines ex...

Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling$fcolon V(D)to {0, 1, 2}$such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ ofRoman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,is called a {em Roman dominating family} (of functions) on $D$....

In mobile ad hoc networks (MANETs), networks can be partitioned into clusters. Clustering algorithms are localized algorithms that have the property of creating an upper bounded clusterheads in any networks even in the worst case. Generally, clusterheads and selected gateways form a connected dominating set (CDS) of the network. This CDS forms the backbone of the network and can be used for rou...

Article history: Received 14 November 2008 Received in revised form 15 January 2012 Accepted 27 February 2012 Available online xxxx

A set S is a 1-movable strong resolving hop dominating of G if for every v ∈ S, either S\{v} or there exists vertex u (V (G)\S)∩NG(v) such that (S \ {v}) ∩ {u} G. The minimum cardinality denoted by γ 1 msRh(G). In this paper, we obtained the corresponding parameter in graphs resulting from join, corona and lexicographic product two graphs. Specifically, characterize sets these types determine b...

In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. After each attack, if the guards can “move” to form a dominating set that contains the attacked vertex, then the guards have successfully defended against the attack. We wish to determine the minimum number of guards required to successfully defend against any possible sequence...