For an integer k ≥,2, a Roman k-tuple dominating function, (or just RkDF), in a graph G is a function f: V (G)→, {0,1,2} satisfying the condition that every vertex u for which f(u) ≠,0 is adjacent to at least k vertices v for which f(v) = 2, and every vertex u for which f(u) 6= 0 is adjacent to at least k-1 vertices v for which f(v) = 2. The Roman k-tuple domination number of G is the minimum weight of an RkDF in G. In this note we settle two problems posed in [Roman k-tuple Domination in Graphs, Iranian J. Math. Sci. Inform. 15 (2020), 101-115].

For any integer k ≥ 1 and any graph G = (V, E) with minimum degree at least k − 1, we define a function f: V → {0, 1, 2} as a Roman k-tuple dominating function on G if for any vertex v with f(v) = 0 there exist at least k and for any vertex v with f(v) ̸ = 0 at least k − 1 vertices w in its neighborhood with f(w) = 2. The minimum weight of a Roman k-tuple dominating function f on G is called the Roman k-tuple domination number of the graph where the weight of f is f(V ) = ∑ v∈ V f(v). In this paper, we initiate to study the Roman k-tuple domination number of a graph, by giving some tight bounds for the Roman k-tuple domination number of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding the Roman k-tuple domination number of some known graphs is our other goal. Some of our results extend these one given by Cockayne and et al. [1] in 2004 for the Roman domination number.

Let k be a positive integer. A subset S of V (G) in a graph G is a k-tuple total dominating set of G if every vertex of G has at least k neighbors in S. The k-tuple total domination number g ´k, t (G) of G is the minimum cardinality of a k-tuple total dominating set of G. In this paper for a given graph G with minimum degree at least k, we find some sharp lower and upper bounds on the k-tuple total domination number of the m-Mycieleskian graph mm (G) of G in terms on k and g´ k, t (G).Specially we give the sharp bounds g´k, t (G) +1 and g´k, t (G) +k for g´ k, t (m1 (G)), and characterize graphs with g´ k, t (m1 (G)) = g´ k, t (G) +1.

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.

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 $G=(V,E)$\ be a simple graph and $f:V\rightarrow\{0,1,2,3\}$ be a function. A vertex $u$ with $f\left( u\right) =0$ is called an undefended vertex with respect to $f$ if it is not adjacent to a vertex $v$ with $f(v)\geq2.$ We call the function $f$ a generous Roman dominating function (GRDF) if for every vertex with $f\left( u\right) =0$ there exists at least a vertex $v$ with $f(v)\geq2$ adjacent to $u$ such that the function $f^{\prime}:V\rightarrow \{0,1,2,3\}$, defined by $f^{\prime}(u)=\alpha$, $f^{\prime}(v)=f(v)-\alpha$ where $\alpha=1$ or $2$, and $f^{\prime}(w)=f(w)$ if $w\in V-\{u,v\}$ has no undefended vertex. The weight of a generous Roman dominating function $f$ is the value $f(V)=\sum_{u\in V}f(u)$. The minimum weight of a generous Roman dominating function on a graph $G$\ is called the generous Roman domination number of $G$, denoted by $\gamma_{gR}\left( G\right) $. In this paper, we initiate the study of generous Roman domination and show its relationships. Also, we give the exact values for paths and cycles. Moreover, we present an upper bound on the generous Roman domination number, and we characterize cubic graphs $G$ of order $n$ with $\gamma_{gR}\left( G\right) =n-1$, and a Nordhaus-Gaddum type inequality for the parameter is also given. Finally, we study the complexity of this parameter.

In this paper, we study the domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph G. We also compute the domination number of some families of graphs such as star graphs, double start graphs, path graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the domination number of middle 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.

A subset $D$ of the vertex set $V(G)$ in a graph $G$ is a point-set dominating set (or, in short, psd-set) of $G$ if for every set $S\subseteq V- D$, there exists a vertex $v\in D$ such that the induced subgraph $\langle S\cup \{v\}\rangle$ is connected.  The minimum cardinality of a psd-set of $G$ is called the point-set domination number of $G$. In this paper, we establish two sharp lower bounds for point-set domination number of a graph in terms of its diameter and girth. We characterize graphs for which lower bound of point set domination number is attained in terms of its diameter. We also establish an upper bound and give some classes of graphs which attains the upper bound of point set domination number.