﻿ Nasrin Dehgardi

# Nasrin Dehgardi

Sirjan University of Technology, Sirjan 78137, Iran

## [ 1 ] - Mixed Roman domination and 2-independence in trees

‎‎Let \$G=(V‎, ‎E)\$ be a simple graph with vertex set \$V\$ and edge set \$E\$‎. ‎A {em mixed Roman dominating function} (MRDF) of \$G\$ is a function \$f:Vcup Erightarrow {0,1,2}\$ satisfying the condition that every element \$xin Vcup E\$ for which \$f(x)=0\$ is adjacent‎‎or incident to at least one element \$yin Vcup E\$ for which \$f(y)=2\$‎. ‎The weight of an‎‎MRDF \$f\$ is \$sum _{xin Vcup E} f(x)\$‎. ‎The mi...

## [ 2 ] - Reformulated F-index of graph operations

The first general Zagreb index is defined as \$M_1^lambda(G)=sum_{vin V(G)}d_{G}(v)^lambda\$. The case \$lambda=3\$, is called F-index. Similarly, reformulated first general Zagreb index is defined in terms of edge-drees as \$EM_1^lambda(G)=sum_{ein E(G)}d_{G}(e)^lambda\$ and the reformulated F-index is \$RF(G)=sum_{ein E(G)}d_{G}(e)^3\$. In this paper, we compute the reformulated F-index for some grap...

## [ 3 ] - Signed total Roman k-domination in directed graphs

Let \$D\$ be a finite and simple digraph with vertex set \$V(D)\$‎.‎A signed total Roman \$k\$-dominating function (STR\$k\$DF) on‎‎\$D\$ is a function \$f:V(D)rightarrow{-1‎, ‎1‎, ‎2}\$ satisfying the conditions‎‎that (i) \$sum_{xin N^{-}(v)}f(x)ge k\$ for each‎‎\$vin V(D)\$‎, ‎where \$N^{-}(v)\$ consists of all vertices of \$D\$ from‎‎which arcs go into \$v\$‎, ‎and (ii) every vertex \$u\$ for which‎‎\$f(u)=-1\$ has a...

## [ 4 ] - The minus k-domination numbers in graphs

For any integer  ‎, ‎a minus  k-dominating function is a‎function  f‎ : ‎V (G)  {-1,0‎, ‎1} satisfying w) for every  vertex v, ‎where N(v) ={u V(G) | uv  E(G)}  and N[v] =N(v)cup {v}. ‎The minimum of ‎the values of  v)‎, ‎taken over all minus‎k-dominating functions f,‎ is called the minus k-domination‎number and is denoted by \$gamma_k^-(G)\$ ‎. ‎In this paper‎, ‎we ‎introduce the study of minu...

## [ 5 ] - Nonnegative signed total Roman domination in graphs

‎Let \$G\$ be a finite and simple graph with vertex set \$V(G)\$‎. ‎A nonnegative signed total Roman dominating function (NNSTRDF) on a‎ ‎graph \$G\$ is a function \$f:V(G)rightarrow{-1‎, ‎1‎, ‎2}\$ satisfying the conditions‎‎that (i) \$sum_{xin N(v)}f(x)ge 0\$ for each‎ ‎\$vin V(G)\$‎, ‎where \$N(v)\$ is the open neighborhood of \$v\$‎, ‎and (ii) every vertex \$u\$ for which‎ ‎\$f(u...

## [ 6 ] - Outer independent Roman domination number of trees

‎A Roman dominating function (RDF) on a graph G=(V,E) is a function  f : V → {0, 1, 2}  such that every vertex u for which f(u)=0 is‎ ‎adjacent to at least one vertex v for which f(v)=2‎. ‎An RDF f is called‎‎an outer independent Roman dominating function (OIRDF) if the set of‎‎vertices assigned a 0 under f is an independent set‎. ‎The weight of an‎‎OIRDF is the sum of its function values over ...

## [ 7 ] - On the outer independent 2-rainbow domination number of Cartesian products of paths and cycles

‎Let G be a graph‎. ‎A 2-rainbow dominating function (or‎ 2-RDF) of G is a function f from V(G)‎ ‎to the set of all subsets of the set {1,2}‎ ‎such that for a vertex v ∈ V (G) with f(v) = ∅, ‎the‎‎condition \$bigcup_{uin N_{G}(v)}f(u)={1,2}\$ is fulfilled‎, wher NG(v)  is the open neighborhood‎‎of v‎. ‎The weight of 2-RDF f of G is the value‎‎\$omega (f):=sum _{vin V(G)}|f(v)|\$‎. ‎The 2-rainbow‎‎d...

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