دوستعلی مژده

A global restrained Roman dominating function on a graph $G=(V,E)$ to be a function $f:V\rightarrow\{0,1,2\}$ such that $f$ is a restrained Roman dominating function of both $G$ and its complement $\overline G$. The weight of a global restrained Roman dominating function is the value $w(f)=\Sigma_{u \in V} f(u)$. The minimum weight of a global restrained Roman dominating function of $G$ is called the global restrained Roman domination number of $G$ and denoted by $\gamma_{grR}(G)$. In this paper we initiate the study of global restrained Roman domination number of graphs. We then prove that the problem of computing $\gamma_{grR}$ is NP-hard even for bipartite and chordal graphs. The global restrained Roman domination of a given graph is studied versus to the other well known domination parameters such as restrained Roman domination number $\gamma_{rR}$ and global domination number $\gamma_g$ by bounding $\gamma_{grR}$ from below and above involving $\gamma_{rR}$ and $\gamma_g$ for general graphs, respectively. We characterize graphs $G$ for which $\gamma_{grR}(G)\in \{1,2,3,4,5\}$. It is shown that: for trees $T$ of order $n$, $\gamma_{grR}(T)=n$ if and only if diameter of $T$ is at most $5$. Finally, the triangle free graphs $G$ for which $\gamma_{grR}(G)=|V|$ are characterized.

A double Italian  dominating (DID) function  of a graph $G=(V,E)$ is a function $f: V(G)\to\{0,1,2,3\}$ havingthe property that for every vertex $v\in V$, $\sum_{u\in N_G[v]}f(u)\geq 3$, if $f(v)\in \{0,1\}$.A restrained  double Italian dominating (RDID) function is a DID function $f$  such that the subgraph induced by the verticeswith label $0$ has no isolated vertex.A total restrained double Italian dominating (TRDID) function is an RDID function $f$  such that the set $\{v\in V: f(v)> 0\}$  induces a subgraph with no isolated vertex.\\We initiate the study of TRDID function of any graph $G$. The TRDID and RDID functions of the middle of any graph $G$ are investigated,and then,  the sharp bounds for these parameters are established.Finally, for  a  graph $H$, we provide the minimum value of TRDID and RDID functions for corona graphs,$H \circ K_1$, $H \circ K_2$ and middle of them.

A proper coloring of a graph G is called a dominated coloring whenever each color class is dominated by at least one vertex. The minimum number of colors among all dominated colorings of G is called its dominated chromatic number, denoted by χ_{dom}(G). We define a parameter related to dominated coloring, namely dominated chromatic covering. For a minimum dominated coloring of G, a set of vertices S is called a dominated chromatic covering if each color class is dominated by a vertex of S. The minimum cardinality of a dominated chromatic covering of G is called its dominated chromatic covering number, denoted by θ_{χ_{dom}}(G). It is clear that θ_{χ_{dom}}(G) ≤ χ_{dom}(G). In this paper, we obtain the dominated chromatic number and θ_{χ_{dom}}(G) when G is middle and total graph of paths and cycles.

Chemical study regarding total $\pi$-electron energy with respect to conjugated molecules has focused on the second Zagreb index of graphs. Moreover, in the last half-century, it has gotten a lot of attention. The relationship between the Roman domination number and the second Zagreb index is investigated in this study. We characterize the trees with the maximum second Zagreb index among those with the given Roman domination number.

For a graph $G$ with no isolated vertex, a covering total double Roman dominating function ($CTDRD$ function) $f$ of $G$ is a total double Roman dominating function ($TDRD$ function) of $G$ for which the set ${vin V(G)| f(v)ne 0}$ is a vertex cover set. The covering total double Roman domination number $gamma_{ctdR}(G)$ equals the minimum weight of an $CTDRD$ function on $G$. An $CTDRD$ function on $G$ with weight $gamma_{ctdR} (G)$ is called a $gamma_{ctdR} (G)$-function. In this paper, the graphs $G$ with small $gamma_{ctdR} (G)$ are characterised. We show that the decision problem associated with $CTDRD$ is $NP$-complete even when restricted to planer graphs with maximum degree at most four. We then show that for every graph $G$ without isolated vertices, $gamma_{oitR}(G)<gamma_{ctdR}(G)< 2gamma_{oitR}(G)$ and for every tree $T$, $2beta(T)+1leq gamma_{ctdR}(T)leq4beta(T)$, where $gamma_{oitR}(G)$ and $beta(T)$ are the outer independent total Roman domination number of $G$, and the minimum vertex cover number of $T$ respectively. Moreover we investigate the $gamma_{ctdR}$ of corona of two graphs.

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set of G. For an eveninteger $nge 2$ and $1le Delta le lfloorlog_2nrfloor$, aKnodel graph $W_{Delta,n}$ is a $Delta$-regularbipartite graph of even order n, with vertices (i,j), for$i=1,2$ and $0le jle n/2-1$, where for every $j$, $0le jlen/2-1$, there is an edge between vertex $(1, j)$ and every vertex$(2,(j+2^k-1)$ mod (n/2)), for $k=0,1,cdots,Delta-1$. In thispaper, we determine the total domination number in $3$-regularKnodel graphs $W_{3,n}$.