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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.

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.

A function $f:V\rightarrow \{0,1,2\}$ on a signed graph $S=(G,\sigma)$  where $G = (V,E)$ is a Roman dominating function(RDF) if $f(N[v]) = f(v) + \sum_{u \in N(v)} \sigma(uv)f(u) \geq 1$ for all $v\in V$ and for each vertex $v$ with $f(v)=0$ there is a vertex $u$ in $N^+(v)$ such that $f(u) = 2$. The weight of an RDF $f$ is given by $\omega(f) =\sum_{v\in V}f(v)$ and the minimum weight among all the RDFs on $S$ is called the Roman domination number $\gamma_R(S)$. Any RDF on $S$ with the minimum weight is known as a $\gamma_R(S)$-function. In this article we obtain certain bounds for $ \gamma_{R} $ and characterise the signed graphs attaining small values for $ \gamma_R. $

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.

Let G = (V; E) be a simple graph. A subset S  V (G) is a dominating set of G if every vertex in V (G) n S is adjacent to at least one vertex in S: The domination number of graph G; denoted by (G); is the minimum size of a dominating set of vertices V (G): Let G1 and G2 be two disjoint copies of graph G and f: V (G1)! V (G2) be a function. Then a functigraph G with function f is denoted by C(G; f); its vertices and edges are V (C(G; f)) = V (G1) [ V (G2) and E(C(G; f)) = E(G1) [ E(G2) [ f vu j v 2 V (G1); u 2 V (G2); f(v) = u g; respectively. In this paper, we investigate domination number of comple-ments of functigraphs. We show that for any connected graph G; (C(G; f)) ⩽ 3: Also we provide conditions for the function f in some graphs such that (C(G; f)) = 3: Finally, we prove if G is a bipartite graph or a connected k regular graph of order n ⩾ 4 for k 2 f 2; 3; 4 g and G = 2 f K3; K4; K5; H1; H2 g; then (C(G; f)) = 2.

‎Let $G=(V‎, ‎E)$ be a simple graph‎. ‎A set $C$ of vertices of $G$ is an identifying code of $G$ if for every two vertices $x$ and $y$ the sets $N_{G}[x] \cap C$ and $N_{G}[y] \cap C$ are non-empty and different‎. ‎Given a graph $G,$ the smallest size of an identifying code of $G$ is called the identifying code number of $G$ and denoted by $\gamma^{ID}(G).$ In this paper‎, ‎we prove that the identifying code number of the subdivision of a graph $G$ of order $n$ is at most $n$‎. ‎Also‎, ‎we prove that the identifying code number of the subdivision of graphs $K_n$, $K_{r,s}$ and $C_P(s)$ are $n‎$,‎ ‎‎‎$‎‎r+s$ and $2s$, respectively‎. ‎Finally‎, ‎we conjecture that for every graph $G$ of order $n$ the identifying code number of the subdivision of $G$ is $n$‎.