# Nasrin SoltankhahAlzahra University · Department of Mathematics

Nasrin Soltankhah

Doctor of Philosophy

## About

43

Publications

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127

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Introduction

## Publications

Publications (43)

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if every vertex $v\in V$ is adjacent to an element of $S$. The domination number of a graph $G$ denoted by $\gamma(G...

A 3-way (v,k,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(v,k,t)$\end{document} trade T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \use...

For any graph $G$ of order $n$ with degree sequence $d_{1}\geq\cdots\geq d_{n}$, we define the double Slater number $s\ell_{\times2}(G)$ as the smallest integer $t$ such that $t+d_{1}+\cdots+d_{t-e}\geq2n-p$ in which $e$ and $p$ are the number of end-vertices and penultimate vertices of $G$, respectively. We show that $\gamma_{\times2}(G)\geq s\ell...

We continue the study of restrained double Roman domination in graphs. For a graph $G=\big{(}V(G),E(G)\big{)}$, a double Roman dominating function $f$ is called a restrained double Roman dominating function (RDRD function) if the subgraph induced by $\{v\in V(G)\mid f(v)=0\}$ has no isolated vertices. The restrained double Roman domination number (...

In this paper, we show that for all $v\equiv 0,1$ (mod 5) and $v \geq 15$, there exists a super-simple $(v,5,2)$ directed design. Also, we show that for these parameters there exists a super-simple $(v,5,2)$ directed design such that its smallest defining sets contain at least half of its blocks.

A 3-way $(v,k,t)$ trade $T$ of volume $m$ consists of three pairwise disjoint collections $T_1$, $T_2$ and $T_3$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$, the number of blocks containing this $t$-subset is the same in each $T_i$ for $1\leq i\leq 3$. If any $t$-subset of found($T$) occurs at most once in each $T...

A 3-way T-trade consists of three disjoint decompositions of some simple graph H without isolated vertices into copies of T. The number of vertices of the graph H is the foundation of the trade, and the number of copies of T in each of the decompositions is the volume of the trade. In this paper, we determine all values v and s for which there exis...

The zero forcing number of a graph is the minimum size of a zero forcing set. This parameter bounds the path cover number which is the minimum number of vertex-disjoint-induced paths that cover all the vertices of graph. In this paper, we investigate these two parameters and present an infinite number of graphs with the large difference between the...

The flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k + r triples, r of them being the triples of a common flower. In this article we determine the set J3F(r) for an...

A u-way (v; k; t) trade is a pair T = (X; T_1; T2,...,T_u) such that for each t-subset of v-set X the number of blocks containing this t-subset is the same in each Ti (1 <= i <=u). In the other words for each 1 <= i < j <= u, (X; T_i; T_j) is a (v; k; t) trade. There are many questions concerning u-way trades. The main question is about the minimum...

A \(\mu \)-way (v, k, t) trade is a pair \(T=(X,\{T_1,T_2,\ldots , T_{\mu }\})\) such that for eacht-subset of v-set X the number of blocks containing this t-subset is the same in each \(T_i\)\((1\le i\le \mu )\). In the other words for each \(1\le i<j\le \mu \), the pair \((X,\{T_i,T_j\})\) is a (v, k, t) trade. A \(\mu \)-way (v, k, t) trade \(T=...

A zero forcing set is a new concept in Graph Theory which was introduced in recent years. In this paper, we investigate the relationship between zero forcing sets and algebraic hyperstructures. To this end, we present some new definitions by considering a zero forcing process on a graph G. These definitions help us analyze the zero forcing process...

In this paper, we expand the research on making relationships between graphs and hyperstructures. First, by considering an arbitrary zero forcing set Z on a graph G and it's forcing process, we define zero forcing roots and zero forcing leaves of one vertex. These new definitions provide the opportunity for us to introduce two commutative hypergrou...

A data center network (DCN) plays a crucial role in data center communications and provides the infrastructure for cloud computing services and data-intensive applications in a cloud data center. Network performance is determined by DCN architecture, whose effective design critically depends on ensuring low cost, high availability, and high scalabi...

Let JR3(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{3}_{R}(v)$$\end{document} denote the set of all integers k such that there exists a collection of three KTS...

A 3-way trade of volume consists of three disjoint collections , and , each of blocks of size , such that for every -subset of -set , the number of blocks containing this -subset is the same in each . If any -subset of found(T) occurs at most once in , then is called 3-way Steiner trade. In this paper the spectrum (that is, the set of allowable vol...

A $\mu$-way $(v, k, t)$ trade $T = \{T_{1} , T_{2}, . . ., T_{\mu} \}$ of volume $m$ consists of $\mu$ disjoint collections $T_{1}, T_{2}, \ldots, T_{\mu}$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$ the number of blocks containing this $t$-subset is the same in each $T_{i}$ (for $1 \leq i \leq \mu$). A $\mu$-way...

Let G=(V(G),E(G)) be a graph, γt(G). Let ooir(G) be the total domination and OO-irredundance number of G, respectively. A total dominating set S of G is called a total perfect codetotal perfect code if every vertex in V(G) is adjacent to exactly one vertex of S. In this paper, we show that if G has a total perfect code, then γt(G)=ooir(G). As a con...

In this paper, we show that for all v\pmod 1 (mod 3), there exists a super-
simple (v, 4, 2) directed design. Also, we show that for these parameters there
exists a super-simple (v, 4, 2) directed design whose each defining set has at
least a half of the blocks.

A $\mu$-way $(v,k,t)$ $trade$ of volume $m$ consists of $\mu$ disjoint
collections $T_1$, $T_2, \dots T_{\mu}$, each of $m$ blocks, such that for
every $t$-subset of $v$-set $V$ the number of blocks containing this t-subset
is the same in each $T_i\ (1\leq i\leq \mu)$. In other words any pair of
collections $\{T_i,T_j\}$, $1\leq i<j \leq \mu$ is a...

A 3-way (v, k, t) trade of volume s consists of 3 disjoint collections and , each of s blocks, such that for every t-subset of v-set V, the number of blocks containing this t-subset is the same in each ( ). In this paper we prove the existence of: (i) 3-way (v, k, 1) trades (Steiner trades) of each volume . (ii) 3-way (v, k, 2) trades of each volum...

In this paper the 3-way intersection problem for $S(2,4,v)$ designs is
investigated. Let $b_{v}=\frac {v(v-1)}{12}$ and
$I_{3}[v]=\{0,1,...,b_{v}\}\setminus\{b_{v}-7,b_{v}-6,b_{v}-5,b_{v}-4,b_{v}-3,b_{v}-2,b_{v}-1\}$.
Let $J_{3}[v]=\{k|$ there exist three $S(2,4,v)$ designs with $k$ same common
blocks$\}$. We show that $J_{3}[v]\subseteq I_{3}[v]$...

Let $K$ be a set of $k$ positive integers. A biclique cover of type $K$ of a
graph $G$ is a collection of complete bipartite subgraphs of $G$ such that for
every edge $e$ of $G$, the number of bicliques need to cover $e$ is a member of
$K$. If $K=\{1,2,..., k\}$ then the maximum number of the vertices of a
complete graph that admits a biclique cove...

A d-biclique cover of a graph G is a collection of bicliques of G such that
each edge of G is in at least d of the bicliques. The number of bicliques in a
minimum d-biclique cover of G is called the d-biclique covering number of G and
is denoted by ${bc}_d(G)$. In this paper, we present an upper bound for the d-
biclique covering number of the lexi...

The intersection problem for a pair of 2-(v, 3, 1) directed designs and 2-(v,
4, 1) directed designs is solved by Fu in 1983 and by Mahmoodian and Soltankhah
in 1996, respectively. In this paper we determine the intersection problem for
2-(v, 5, 1) directed designs.

A $2-(v,k,\lambda)$ directed design (or simply a $2-(v,k,\lambda)DD$) is
super-simple if its underlying $2-(v,k,2\lambda)BIBD$ is super-simple, that is,
any two blocks of the $BIBD$ intersect in at most two points. A
$2-(v,k,\lambda)DD$ is simple if its underlying $2-(v,k,2\lambda)BIBD$ is
simple, that is, it has no repeated blocks.
A set of blocks...

In this paper we investigate the spectrum of super-simple 2-$(v,5,1)$
directed designs (or simply super-simple 2-$(v,5,1)$DDs) and also the size of
their smallest defining sets.
We show that for all $v\equiv1,5\ ({\rm mod}\ 10)$ except $v=5,15$ there
exists a super-simple $(v,5,1)DD$. Also for these parameters, except possibly
$v=11,91$, there exis...

A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v∈V is adjacent to an element of S. A set S of vertices in a graph G(V,E) is called a total restrained dominating set if every vertex v∈V is adjacent to an element of S and every vertex of V-S is adjacent to a vertex in V-S. The total domination number of a graph...

A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v∈V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Total domination subdivision number denoted by sdγt is the minimum number of edges that must be subdivided to...

Mahmoodian and Soltankhah [6] conjectured that there does not exist any t-(v,k) trade of volume si < s < Si+1, where S i = 2t+1 - 2t-i, i = O, 1,..., t - 1. Also they showed that the conjecture is true for i = O. In this paper we prove the correctness of this conjecture for Steiner trades.

Mahmoodian and Soltankhah $\cite{MMS}$ conjectured that there does not exist any $t$-$(v,k)$ trade of volume $s_{i}< s <s_{i+1}$, where $s_{i}=2^{t+1}-2^{t-i}, i=0,1,..., t-1$. Also they showed that the conjecture is true for $i=0$. In this paper we prove the correctness of this conjecture for Steiner trades.

In a given graph $G$, a set $S$ of vertices with an assignment of colors is a {\sf defining set of the vertex coloring of $G$}, if there exists a unique extension of the colors of $S$ to a $\Cchi(G)$-coloring of the vertices of $G$. A defining set with minimum cardinality is called a {\sf smallest defining set} (of vertex coloring) and its cardinal...

Let d (sigma) stand for the defining number of the colouring sigma. In this paper we consider drain = min, d(y) and d(max) = max gamma d(gamma) for the onto chi-colourings gamma of the circular complete graph K-n,K-d. In this regard we obtain a lower bound for d(min)(K-n,K-d) and we also prove that this parameter is asymptotically equal to chi-1. A...

In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is...

In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is...

A t-(v,k,λ) directed design (or simply a t-(v,k, λ)DD) is a pair (V,B), where V is a v-set and B is a collection of (transitively) ordered k-tuples of distinct elements of V, such that every ordered t-tuple of distinct elements of V belongs to exactly λ elements of B. (We say that a t-tuple belongs to a k-tuple, if its components are contained in t...

The concept of defining set has been studied in block designs and, under the name critical sets, in Latin squares and Room squares. Here we study defining sets for directed designs. A t-(v, k, λ) directed design (DD) is a pair (V, B), where V is a v-set and B is a collection of ordered blocks (or k-tuples of V), for which each t-tuple of V appears...

A t-(v, k, λ) directed design (or simply a t-(v, k, λ)DD) is a pair (V, B), where V is a v-set and B is a collection of ordered k-tuples of distinct elements of V, such that every ordered t-tuple of distinct elements of V belongs to exactly λ elements of B. (We say that a t-tuple belongs to a k-tuple, if its components appear in that k-tuple as a s...