Nasrin Soltankhah

• Doctor of Philosophy
• Professor (Full) at Alzahra University

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...

فراخوان پسا دکتری

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars attain the bound, and among connected unicyclic graphs such graphs are $t$-cycles for $t\in \{3,4,5\}$. It is shown...

The problem of injective coloring in graphs can be revisited through two different approaches: coloring the two-step graphs and vertex partitioning of graphs into open packing sets, each of which is equivalent to the injective coloring problem itself. Taking these facts into account, we observe that the injective coloring lies between graph colorin...

A $k$-limited packing partition ($k$LP partition) of a graph $G$ is a partition of $V(G)$ into $k$-limited packing sets. We consider the $k$LP partitions with minimum cardinality (with emphasis on $k=2$). The minimum cardinality is called $k$LP partition number of $G$ and denoted by $\chi_{\times k}(G)$. This problem is the dual problem of $k$-tupl...

In this paper, we introduce a connection between two classical concepts of graph theory: metric dimension and distinguishing number. For a given graph G, let dim(G) and D(G) represent its metric dimension and distinguishing number, respectively. We show that in connected graphs, any resolving set breaks the symmetry in the graphs. Precisely, if G i...

In this paper, we show that for all \(v\equiv 0,1\) (mod 5) and \(v\ge 15\), there exists a super-simple (v, 5, 2) directed design. Moreover, 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. Also, we show that these designs are useful in constructin...

A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the distinguishing coloring of a graph and its applications in computing the distinguishing chromatic number of disconnec...

In this paper, we study the fourth weight of generalized Reed–Muller codes. Erickson in his Ph.D. thesis proved that the second weight of \(R_q(a(q-1)+b,m)\) depends on the second weight \(R_q(b,2)\). Also, Leducq (Discret Math 338:1515–1535, 2015) proved that under the same condition, by the third weight of \(R_q(b,2)\) we can determine the third...

Let $G=(V(G),E(G))$ be a graph. A function $f:V(G)\rightarrow \mathbb{P}(\{1,2\})$ is a $2$-rainbow dominating function if for every vertex $v$ with $f(v)=\emptyset$, $f\big{(}N(v)\big{)}=\{1,2\}$. An outer-independent $2$-rainbow dominating function (OI$2$RD function) of $G$ is a $2$-rainbow dominating function $f$ for which the set of all $v\in V...

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

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 fou...

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...

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...