A dominating set DÍV of a graph G= (V; E) is said to be a CONNECTED cototal dominating set if (D) is CONNECTED and (V-D) ¹f, contains no isolated vertices. A CONNECTED cototal dominating set is said to be minimal if no proper subset of D is CONNECTED cototal dominating set. The CONNECTED cototal DOMINATION NUMBER (G) of G is the minimum cardinality of a minimal CONNECTED cototal dominating set of G. In this paper, we begin an investigation of CONNECTED cototal DOMINATION NUMBER and obtain some interesting results.

A $k$-CEC graph is a graph $G$ which has CONNECTED DOMINATION NUMBER $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. A $k$-CVC graph $G$ is a $2$-CONNECTED graph with  $\gamma_{c}(G) = k$ and $\gamma_{c}(G - v) < k$ for any $v \in V(G)$. A graph is said to be maximal $k$-CVC if it is both $k$-CEC and $k$-CVC. Let $\delta$, $\kappa$, and $\alpha$ be the minimum degree, connectivity, and independence NUMBER of $G$, respectively. In this work, we prove that for a maximal $3$-CVC graph, if $\alpha = \kappa$, then $\kappa = \delta$. We additionally consider the class of maximal $3$-CVC graphs with $\alpha < \kappa$ and $\kappa < \delta$, and prove that every $3$-CONNECTED maximal $3$-CVC graph when $\kappa < \delta$ is Hamiltonian CONNECTED.

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.

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.

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.

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.