Let G be a graph with vertex set V . A set D ⊆ V is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in V \D has a neighbor in V \D. The minimum cardinality of a total restrained dominating set of G is called the total restrained domination number of G, and is denoted by γtr(G). In this paper, we prove that if G is a connected graph of order n ≥ 4...

Given graph G = (V,E), a dominating set S is a subset of vertex set V such that any vertex not in S is adjacent to at least one vertex in S. The domination number of a graph G is the minimum size of the dominating sets of G. In this paper we study some results on domination number, connected, independent, total and restrained domination number denoted by γ(G), γc(G) ,γi(G), γt(G) and γr(G) resp...

The following fundamental result for the domination number γ(G) of a graph G was proved by Alon and Spencer, Arnautov, Lovász and Payan: γ(G) ≤ ln(δ + 1) + 1 δ + 1 n, where n is the order and δ is the minimum degree of vertices of G. A similar upper bound for the double domination number was found by Harant and Henning [On double domination in graphs. Discuss. Math. Graph Theory 25 (2005) 29–34...

A {em Roman dominating function} on a graph $G$ is a function$f:V(G)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}A {em restrained Roman dominating}function} $f$ is a {color{blue} Roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} The wei...

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |NG[v]∩S| ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V , |NG(v)∩S| ≥ k. The k-transversal numb...

Let D be a minimum secure restrained dominating set of a graph G = (V, E). If V – D contains a restrained dominating set D' of G, then D' is called an inverse restrained dominating set with respect to D. The inverse restrained domination number γr(G) of G is the minimum cardinality of an inverse restrained dominating set of G. The disjoint restrained domination number γrγr(G) of G is the minimu...

let $r$ be a commutative ring and $m$ be an $r$-module with $t(m)$ as subset,   the set of torsion elements. the total graph of the module denoted   by $t(gamma(m))$, is the (undirected) graph with all elements of   $m$ as vertices, and for distinct elements  $n,m in m$, the   vertices $n$ and $m$ are adjacent if and only if $n+m in t(m)$. in this paper we   study the domination number of $t(ga...

In this paper, we provide a new upper bound for the α-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the α-rate domination number, which combines the concepts of...