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For any integer k³1, a set S of vertices in a graph G=(V,E) is a k-tuple total dominating set of G if any vertex of G is adjacent to at least k vertices in S, and any vertex of V-S is adjacent to at least k vertices in V-S. The minimum NUMBER of vertices of such a set in G we call the k-tuple total restrained DOMINATION NUMBER of G. The maximum NUMBER of classes of a partition of V such that its all classes are k-tuple total restrained dominating sets in G we call the k-tuple total restrained DOMATIC NUMBER of G.In this paper, we give some sharp bounds for the k-tuple total restrained DOMINATION NUMBER of a graph, and also calculate it for some of the known graphs. Next, we mainly present basic properties of the k-tuple total restrained DOMATIC NUMBER of a graph.

For a graph G = (V; E), a partition  = fV1; V2; : : :; Vkg of the vertex set V is an upper DOMATIC partition if Vi dominates Vj or Vj dominates Vi or both for every Vi; Vj 2  , whenever i 6= j. The upper DOMATIC NUMBER D(G) is the maximum order of an upper DOMATIC partition of G. We study the properties of upper DOMATIC NUMBER and propose an upper bound in terms of clique NUMBER. Further, we discuss the upper DOMATIC NUMBER of certain graph classes including unicyclic graphs and power graphs of paths and cycles.

Let A and B be two disjoint subsets of the vertex set V of a graph G. The set A is said to dominate B, denoted by A! B, if for every vertex u 2 B there exists a vertex v 2 A such that uv 2 E(G). For any graph G, a partition  = fV1; V2; : : :; Vpg of the vertex set V is an upper DOMATIC partition if Vi! Vj or Vj! Vi or both for every Vi; Vj 2  , whenever i 6= j. The upper DOMATIC NUMBER D(G) is the maximum order of an upper DOMATIC partition. In this paper, we study the upper DOMATIC NUMBER of powers of graphs and examine the special case when power is 2. We also show that the upper DOMATIC NUMBER of k th power of a graph can be viewed as its k-upper DOMATIC NUMBER.

In this paper, we investigate DOMINATION NUMBER as well as signed DOMINATION NUMBERs of Cay(G: S) for all cyclic group G of order n, where n ϵ {pm, pq} and S = {k < n: gcd(k, n) = 1}. We also introduce some families of connected regular graphs 􀀀 such that S (􀀀 ) ϵ {2, 3, 4, 5}.

Let k  1 be an integer. A weak signed Roman k-dominating function on a graph G is a function f: V (G)! f1; 1; 2g such that P u2N[v] f(u)  k for every v 2 V (G), where N[v] is the closed neighborhood of v. A set ff1; f2; : : :; fdg of distinct weak signed Roman k-dominating functions on G with the property that Pd i=1 fi(v)  k for each v 2 V (G), is called a weak signed Roman k-dominating family (of functions) on G. The maximum NUMBER of functions in a weak signed Roman kdominating family on G is the weak signed Roman k-DOMATIC NUMBER of G, denoted by d k wsR (G). In this paper we initiate the study of the weak signed Roman k-DOMATIC NUMBER in graphs, and we present sharp bounds for d (G). In addition, we determine the weak signed Roman k-DOMATIC NUMBER of some graphs.

A Roman dominating function on a digraph D with vertex set V (D) is a labeling f: V (D)! f0; 1; 2g such that every vertex with label 0 has an in-neighbor with label 2. A set ff1; f2; : : :; fdg of Roman dominating functions on D with the property that Pd i=1 fi(v)  2 for each v 2 V (D), is called a Roman dominating family (of functions) on D. The maximum NUMBER of functions in a Roman dominating family on D is the Roman DOMATIC NUMBER of D, denoted by dR(D). In this note, we study the Roman DOMATIC NUMBER in digraphs, and we present some sharp bounds for dR(D). In addition, we determine the Roman DOMATIC NUMBER of some digraphs. Some of our results are extensions of well-known properties of the Roman DOMATIC NUMBER of undirected graphs.

For any kÎN, the k -subdivision of a graph G is a simple graph G 1/k , which is constructed by replacing each edge of G with a path of length k. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No.10-11, 1551-1556] the m th power of the n-subdivision of G has been introduced as a fractional power of G, denoted by G m/n. In this regard, we investigate DOMINATION NUMBER and independent DOMINATION NUMBER of fractional powers of graphs.