جستجوی مقالات مرتبط با کلیدواژه « Domination Number » در نشریات گروه « علوم پایه »

Let $G=(V,E)$ be a connected graph. A subset $S$ of $V$ is called a $\gamma$-free set if there exists a $\gamma$-set $D$ of $G$ such that $S \cap D= \emptyset$. If further the induced subgraph $H=G[V-S]$ is connected, then $S$ is called a  $cc$-$\gamma$-free set of $G$. We use this concept to identify connected induced subgraphs $H$ of a given graph $G$ such that $\gamma(H) \leq \gamma(G)$. We also introduce the concept of $\gamma$-totally-free and $\gamma$-fixed sets and present several basic results on the corresponding parameters.

The comaximal intersection graph $CI(R)$ of ideals of a ring $R$ is an undirected graph whose vertex set is the collection of all non-trivial (left) ideals of $R$ and any two vertices $I$ and  $J$ are adjacent if and only if $I+J=R$ and $I\cap J\neq0$. We study the connectedness of $CI(R)$. We also discuss  independence number, clique number, domination number, chromatic number of $CI(R)$.

The concept of (quasi) superhypergraphs as a generalization of graphs makes a relation between some sets of elements in detail and in general (in the form of parts to parts, parts to whole, and whole to whole elements of sets) and is very useful in the real world. This paper considers the novel concept of (quasi) superhypergraphs and introduces the notation of dominating set and domination number of (quasi) superhypergraphs. Especially, we have analyzed the domination number of uniform (quasi) superhypergraphs and computed their domination number on different cases. The flows (from right to left, from left to right, and two-sided) as maps play a main role in (quasi) superhypergraphs and it is proved that domination numbers of (quasi) superhypergraphs are dependent on the flows. We define the valued-star (quasi) superhypergraphs for the design of hypernetworks and compute their domination numbers. We have shown that the domination numbers of valued-star (quasi) superhypergraphs are distinct in different flow states. In final, we introduce some applications of dominating sets of (quasi) superhypergraphs in hypernetwork as computer networks and treatment networks with the optimal application.

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.

Let $R$ be an associative ring and $M$ be a monoid‎. ‎In this paper‎, ‎we introduce new kind of graph structure asociated with zero-divisors of monoid ring $R[M]$‎, ‎calling it the $M$-Armendariz graph of a ring $R$ and denoted by $A(R,M)$‎. ‎It is an undirected graph whose vertices are all non-zero zero-divisors of the monoid ring $R[M]$ and two distinct vertices $\alpha=a_{1}g_{1}+\cdots‎+ ‎a_{n}g_{n}$ and $\beta=b_{1}h_{1}+\cdots+b_{m}h_{m}$ are adjacent if and only if $a_{i}b_{j}=0$ or $b_{j}a_{i}=0$ for all $i,j$‎. ‎We investigate some graph properties of $A(R,M)$ such as diameter‎, ‎girth‎, ‎domination number and planarity‎. ‎Also‎, ‎we get some relations between diameters of the $M$-Armendariz graph $A(R,M)$ and that of zero divisor graph $\Gamma(R[M])$‎, ‎where $R$ is a reversible ring and $M$ is a unique product monoid‎.

Let $G_{\tau }=\left(V,\ E\right)$ be a topological graph which is a finite, simple, undirected, connected graph without isolated vertices. In this paper, several bounds and domination parameters are studied and applied to it: bi-domination, doubly connected bi-domination and pitchfork domination. The dominating set and domination number with its inverse for all these types are calculated. Also, some figures from the topological graph are introduced.

‎Consider a ring $R$ with order $p$ or $p^2$‎, ‎and let $\mathcal{P}(R)$ represent its multiplicative power graph‎. ‎For two distinct rings $R_1$ and $R_2$ that possess identity element 1‎, ‎we define a new structure called the unit semi-cartesian product of their multiplicative power graphs‎. ‎This combined structure‎, ‎denoted as $G.H$‎, ‎is constructed by taking the Cartesian product of the vertex sets $V(G) \times V(H)$‎, ‎where $G = \mathcal{P}(R1)$ and $H = \mathcal{P}(R2)$‎. ‎The edges in $G.H$ are formed based on specific conditions‎: ‎for vertices $(g,h)$ and $(g^\prime,h^\prime)$‎, ‎an edge exists between them if $g = g^\prime$‎, ‎$g$ is a vertex in $G$‎, ‎and the product $hh^\prime$ forms a vertex in $H$‎. ‎Our exploration focuses on understanding the characteristics of the multiplicative power graph resulting from the unit semi-cartesian product $\mathcal{P}(R1).\mathcal{P}(R2)$‎, ‎where $R_1$ and $R_2$ represent distinct rings‎. ‎Additionally‎, ‎we offer insights into the properties of the multiplicative power graphs inherent in rings of order $p$ or $p^2$‎.

In communication networks, strong connectivity between nodes is critical. The failure of strong connectivity between nodes may jeopardize the network’s stability. In fuzzy graphs, various dominating sets using strong edges are identified to avoid network stability. In this paper, the concept of bridge domination set and bridge domination number  in fuzzy graphs is introduced. A few prominent properties of bridge domination numbers are chosen and analyzed using relevant examples. The bridge domination number of fuzzy trees, constant fuzzy cycles, and complete fuzzy and bipartite fuzzy graphs are identified. The use of bridge domination in a partial mesh topology to ensure network continuity is demonstrated in the event of a node failure.

Let $G_{\tau}=(V, E)$ be a topological graph constructed from the topological space $(X, \tau)$. In this paper, several types of dominating parameters are applied on the topological graph $G_{\tau}$. Such as independent domination, total domination, connected domination, doubly connected domination, restrained domination, strong domination and weak domination. Also, the inverse domination of all these parameters was proved.

In this paper, some graph parameters of the zero-divisor graph $\Gamma(R)$ of a finite commutative ring $R$ for $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{p^2}$ and $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{2p}$ where $p>2$ a prime, are investigated. The graph $\Gamma(R)$ is a simple graph whose vertex set is the set of non-zero zero-divisors of a commutative ring $R$ with non-zero identity and two vertices $u$ and $v$ are adjacent if and only if $uv=vu=0$.
In this paper, we study some of the topological indices such as graph energies, the Zagreb indices and the domination parameters of graphs $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{p^2} \big)$ and $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{2p}\big)$.

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. An {outer-independent total $2$-rainbow dominating function of a graph $G$ is a function $f$ from $V(G)$ to the set of all subsets of ${1,2}$ such that the following conditions hold: (i) for any vertex $v$ with $f(v)=emptyset$ we have $bigcup_{uin N_G(v)} f(u)={1,2}$, (ii) the set of all vertices $vin V(G)$ with $f(v)=emptyset$ is independent and (iii) ${vmid f(v)neqemptyset}$ has no isolated vertex. The outer-independent total $2$-rainbow domination number of $G$, denoted by ${gamma}_{oitr2}(G)$, is the minimum value of $omega(f)=sum_{vin V(G)} |f(v)|$ over all such functions $f$. In this paper, we study the outer-independent total $2$-rainbow domination number of $G$ and classify all graphs with outer-independent total $2$-ainbow domination number belonging to the set ${2,3,n}$. Among other results, we present some sharp bounds concerning the invariant.

A dominating set D of a graph G = (V, E) is a nonsplitdominating set if the induced graph hV − Di is connected. The nonsplitdomination number γns(G) is the minimum cardinality of a nonsplitdomination set. The purpose of this paper is to initiate the investigationof those graphs which are critical in the following sense: A graph G iscalled vertex domination critical if γ(G − v) < γ(G) for every vertex vin G. A graph G is called vertex nonsplit critical if γns(G −v) < γns(G)for every vertex v in G. Initially we test whether some particular classesof graph are γns-critical or not and then we have shown that thereis no existence of 2-γns-critical graph. Then 3-γns-critical graphs arecharacterized.

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.

A weakly completely prime graph of a near ring (WI(N)) was defined in this paper, the relationship between the elements of a near ring was determined by a definition of weakly completely prime ideal, so we studied many concepts related to the statements of the given near ring. We also found some theories related to previous graphs of our papers.

Let S be a semiring with identity and U be a unitary left S-semimodule. The co-intersection graph of an S-semimodule U, denoted by Γ(U), is defined to be the undirected simple graph whose vertices are in one-to-one correspondence with all non-trivial subsemimodules of U, and there is an edge between two distinct vertices N and L if and only if N+L≠U. We study these graphs to relate the combinatorial properties of Γ(U) to the algebraic properties of the S-semimodule U. We study the connectedness of Γ(U). We investigate some properties of Γ(U) for instance, we find the domination number and clique number of Γ(U). Also, we study cycles in Γ(U).

Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R/H is a saturated multiplicatively closed subset of R. Anderson and Badawi [4] introduced the generalized total graph of R as an undirected simple graph GTH(R) with vertex set as R and any two distinct vertices x and y are adjacent if and only if x + y ϵ H. The main objective of this paper is to study the domination properties of the graph GTH(R). We determine the domination number of GTH(R) and its induced subgraphs GTH(H) and GTH(R/H). We establish a relationship between the domination number of GTH(R) and the same of GTH(R/H). We also establish a relationship between diameter and domination number of GTH(R/H). In addition,we obtain the bondage number of GTH(R). Finally, a relationship between girth and bondage number of GTH(R/H) has been established.

The edge neighborhood graph N_{e}(G) of a simple graph G is the graph with the vertex set E ∪ S where S is the set of all open edge neighborhood sets of G and two vertices u,v ∈ V (N_e{}(G)) adjacent if u ∈ E and v is an open edge neighborhood set containing u. In this paper, we determine the domination number, the total domination number, the independent domination number and the 2-domination number in the edge neighborhood graph. We also obtain a 2-domination polynomial of the edge neighborhood graph for some certain graphs.

For a simple, undirected, connected graph G=(V,E), a function f : V(G) →{0, 1, 2} which satisfies the following conditions is called a quasi-total Roman dominating function (QTRDF) of G with weight f(V(G))=ΣvΕV(G) f(v).C1). Every vertex uεV for which f(u) = 0 must be adjacent to at least one vertex v with f(v) = 2, and C2). Every vertex uεV for which f(u) = 2 must be adjacent to at least one vertex v with f(v)≥1.  For a graph G, the smallest possible weight of a QTRDF of G denoted γqtR(G) is known as the quasi-total Roman domination number of G. The problem of determining γqtR(G) of a graph G is called minimum quasi-total Roman domination problem (MQTRDP). In this paper, we show that the problem of determining whether G has a QTRDF of weight at most l is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. On the positive side, we show that MQTRDP for threshold graphs, chain graphs and bounded treewidth graphs is linear time solvable. Finally, an integer linear programming formulation for MQTRDP is presented.

In this paper, we initiate the study of total outer-convex domination as a new variant of graph domination and we show the close relationship that exists between this novel parameter and other domination parameters of a graph such as total domination, convex domination, and outer-convex domination. Furthermore, we obtain general bounds of total outer-convex domination number and, for some particular families of graphs, we obtain closed formulas.

We study Beck-like coloring of measurable functions on a measure space Ω taking values in a measurable semigroup ∆. To any measure space Ω and any measurable semigroup ∆, we assign a graph (called a zero-divisor graph) whose vertices are labeled by the classes of measurable functions defined on Ω and having values in ∆, with two vertices f and g adjacent if f · g = 0 a.e.. We show that, if Ω is atomic, then not only the Beck’s conjecture holds but also the domination number coincides to the clique number and chromatic number as well. We also determine some other graph properties of such a graph.