Domination Number Research Articles

Characterizations of Minimal Dominating Sets in γ-Endowed and Symmetric γ-Endowed Graphs with Applications to Structure-Property Modeling

Claude Berge (1987) introduced the concept of k-extendable graphs, wherein any independent set of size k is inherently a constituent of a maximum independent set within a graph H=(V,E). Graphs possessing the property of being 1-extendable are termedas Berge graphs. This introduction gave rise to the notion of well-covered graphs and well-dominated graphs. A graph is categorized as well-covered if each of its maximal independent sets is, in fact, a maximum independent set. Similarly, a graph attains the classification of well-dominated if every minimal dominating set (DS) within it is a minimum dominating set. In alignment with the concept of k-extendable graphs, the framework of (k,γ)-endowed graphs and symmetric (k,γ)-endowed graphs are established. In these graphs, each DS of size k encompasses a minimum DS of the graph. In this article, a study of γ-endowed dominating sets is initiated. Various results providing a deep insight into γ-endowed dominating sets in graphs such as those characterizing the ones possessing a unique minimum DS are proven. We also introduce and study the symmetric γ-endowed graphs and minimality of dominating sets in them. In addition, we give a solution to an open problem in the literature. which seeks to find a domination-based parameter that has a correlation coefficient of ρ>0.9967 with the total π-electronic energy of lower benzenoid hydrocarbons. We show that the upper dominating number Γ(H) studied in this paper delivers a strong prediction potential.

On the Strong Secure Domination Number of a Graph

In this paper, we characterize trees with a strong secure domination number less than or equal to 4 and compute this parameter for certain classes of graphs. Also, we investigate bounds for the strong secure domination number of vertex gluing of two graphs.

The Regular Domination Number of Some Special Graphs

The Regular Domination Number of Some Special Graphs

Prime Spectrum Graph of C-Lattices

Let £ be a C-lattice. Let σ(£) be the set of all prime elements of £ and M(£) be the collection of all maximal elements of £. For S ⊆M(£), we introduce the new graph called S-join graph on σ(£), denoted by ΓS(σ(£)). We have studied properties like connectivity, diameter and domination number of the graph σ(£) . In this paper, we established that the topological space σ(£) is connected if and only if the graph ΓS(σ(£)) is connected.

A Study on Optimizing the Maximal Product in Cubic Fuzzy Graphs for Multifaceted Applications

Graphs in the field of science and technology make considerable use of theoretical concepts. When dealing with numerous links and circumstances in which there are varying degrees of ambiguity or robustness in the connections between aspects, rather than purely binary interactions, cubic fuzzy graphs (CFGs) are more adaptable and compatible than fuzzy graphs. To better represent the complexity of interactions or linkages in the real world, an emerging CFG can be very helpful in achieving better problem-solving abilities that specialize in domains like network analysis, the social sciences, information retrieval, and decision support systems. This idea can be used for a variety of uncertainty-related issues and assist decision-makers in selecting the best course of action through the use of a CFG. Enhancing the maximized network of three cubic fuzzy graphs’ decision-making efficiency was the ultimate objective of this study. We introduced the maximal product of three cubic fuzzy graphs to investigate how interval-valued fuzzy membership, fuzzy membership, and the miscellany of relations are all simultaneously supported through the aspect of degree and total degree of a vertex. Furthermore, the domination on the maximal product of three CFGs was illustrated to analyze the minimum domination number of the weighted CFG, and the proposed approach is illustrated with applications.

Split Regular Domination in Litact Graphs

Objectives: In the context of graph theory, a litact graph is a specific type of graph. This study introduces a new domination parameter, called split regular domination in litact graphs. Methods: When we talk about split regular domination in a litact graph during this investigation, we think about how to divide the litact graph into partitions that adhere to specific domination principles by taking a minimal split regular dominating set with all vertices of equal degree. We used a few common definitions and the ideas of several domination parameters in G to obtain the results. Findings: Numerous bounds on were found in relation to the different parameters of G like vertices, edges, diameter, vertex covering number, maximum degree and so forth, and its relationship to other dominating parameters of G such as total domination, edge domination, connected domination and so on was also found. Furthermore, outcomes resembling those of Nordhaus-Gaddum were also obtained. Novelty: Graph G was used to find a litact graph. Subsequently, a few findings of a new domination parameter called split regular domination in a litact graph in terms of different parameters of G have been established. Keywords: Graph, Litact Graph, Split Domination Number, Regular Domination Number, Split Regular Domination Number

Strong Regular Domination in Litact Graphs

Strong regular domination in a litact graph is a novel domination parameter that has been introduced in this paper. A dominating set D⊆V(G) is known as Strong regular dominating set of G, if for each point x∈V(G)-D there is a vertex y∈D with an edge xy∈E(G) and deg⁡(x)≤deg⁡(y) and all vertices of 〈D〉 holds the equal degree. The lowest cardinality of such vertices of D is known as strong regular domination number of G which is represented by γ_str (G). The current study aims by taking strong regular domination on a litact graph m(G) denoted by γ_str [m(G)] and to obtain some bounds on γ_str [m(G)] in terms of various parameters of G such as vertices, edges, maximum degree, diameter and so on and also in terms of various domination parameters of G such as total domination of G, connected domination of G and so on. Furthermore, outcomes resembling those of Nordhaus-Gaddum were also obtained.

Order Structured Graphs of Cyclic Groups and their Classification

Let $\Gamma^{o}(G)$ with $G\cong C_{p},$ a cyclic group of order $p,$ be an order structured graph. The group $C_{p}$ will be assumed as the vertex set of the graph $\Gamma^{o}(G)$ and an edge between vertices will be built on the basis of a defined relation via order structure. Certain graphical parameters such as independence ratio, clique number, domination number, and separability are discussed. Some characterizations are proposed and proved by incorporating the defined relation. It is further proved that $\Gamma^{o}(C_{p})$ can never be a hamiltonian graph. Lastly, It is shown that $C(\Gamma^{o}(C_{p}))$ is isomorphic to $\Gamma^{o}(C_{p}).

A novel local search approach with connected dominating degree-based incremental neighborhood evaluation for the minimum 2-connected dominating set problem

A novel local search approach with connected dominating degree-based incremental neighborhood evaluation for the minimum 2-connected dominating set problem

Power Domination in Generalized Mycielskian of Cycles

Let $ G=G(V,E) $ be a graph. A set $ S\subseteq V $ is a power dominating set of $ G $ if $ S $ observes all the vertices in $ V,$ following two rules: domination and propagation. The cardinality of a minimum power dominating set is called the power domination number. In this paper, we compute the power domination number of generalized $ m $-Mycielskian of a cycle, $ C_n $. We found that it depends on the numbers $ n \an m $.

Some New Results on Domination and Independent Dominating Set of Some Graphs

One area of graph theory that has been studied in great detail is dominance in graphs. Applications for dominating sets are numerous. In wireless networking, dominant sets are used to find effective paths inside ad hoc mobile networks. They have also been used in the creation of document summaries and safe electrical grid systems. A set <I>S</I>⊆<I>V</I> is said to be dominating set of <I>G</I> if for every <i>v </i>є <I>V</I>-<I>S</I> there exists a vertex <i>u</i> є <I>S</I> such that <i>uv</i> є <I>E</I>. The dominance number of <I>G</I>, represented by <i>γ</i>(<I>G</I>), is the lowest cardinality of vertices among the dominating set of <I>G</I>. A classic NP-complete decision problem in computational complexity theory determines whether, given a graph <I>G</I> and input <I>K</I>, <i>γ</i>(<I>G</I>) ≤ <I>K</I>. This is known as the dominating set issue. Consequently, it is thought that calculating <i>γ</i>(<I>G</I>) for each given graph <I>G</I> may not be possible to do with a feasible algorithm. In addition to efficient approximation tactics, there exist efficient exact techniques for various graph classes. If there are no neighboring vertices in a subset <I>S</I>, then <I>S</I>⊆<I>V</I> is an independent set. Additionally, the empty set and the subset with just one vertex are independent. An independent dominating set of <I>G</I> is a set <I>S</I> of vertices in a graph <I>G</I> that is both an independent and a dominating set of <I>G</I>. This paper's primary goal is to investigate the dominance and independent dominating set of many graphs, including the line graph, the alternate triangular belt graph, the bistar graph, the triangular snake graph, and others.

Why English? Exploring Chinese early career returnee academics’ motivations for writing and publishing in English

To improve their research performance in international league tables, many universities in non-English language dominant settings recruit academic returnees in the hope that they will increase the quantity and quality of articles published in English-medium internationally indexed journals. This study explores Chinese early career returnee academics’ motivations for writing and publishing in English. Utilising ecological systems theory, the findings show that the microsystem is reflected in the early career returnee academics’ interaction with collaborators, while national policies constitute the exosystem. The academic culture has a noteworthy impact at the macrosystem level. This study contributes to the understanding of early career returnee academics’ motivations to write and publish in English which will assist policymakers and university administrators to create a more beneficial environment to promote the accomplishments of academic returnees.

Leaf sector covers with applications on circle graphs

Damian and Pemmaraju (2002) [8] introduced leaf sector covers for circle graphs and presented an O(n2)-time algorithm to find this data structure. They subsequently employed it as an algorithmic tool, leading to the development of an 8-approximation algorithm for the dominating set problem, a (2+ε)-approximation scheme for the dominating set problem, and a (3+ε)-approximation scheme for the total dominating set problem.In this paper, we have improved the time complexity from O(n2) to O(n+m) for finding a leaf sector cover. Furthermore, we employed leaf sector covers as a powerful algorithmic tool to design a 10-approximation algorithm for the total dominating set problem and a 12-approximation algorithm for the paired-dominating set problem. Moreover, we also proposed a (2+ε)-approximation scheme for the total dominating set problem and a (4+ε)-approximation scheme for the paired-dominating set problem. The results presented above are currently the best algorithms for circle graphs.

Non-torsion element graph of a module over a commutative ring

Let R be a commutative ring with unity and M be a unitary R-module. Let T(M) be the set of all torsion elements of M and NT(M) = M − T(M) be the set of all non-torsion elements of M. The non-torsion element graph of M over R is an undirected simple graph GNT (M) with NT(M) as vertex set and any two distinct vertices x and y are adjacent if and only if x + y ∈ T(M). In this paper, we study the basic properties of the graph GNT (M). We also study the diameter and girth of GNT (M). Further, we determine the domination number and the bondage number of GNT (M). We establish a relation between diameter and domination number of GNT (M). We also establish a relation between girth and bondage number of GNT (M). We also establish a relation between girth and bondage number of GNT (M).

Gated independence in graphs

If G=(V,E) is a (finite and simple) graph, we call an independent set X a gated independent set in G if for each x∈X, there exists a neighbor y of x such that (X∖{x})∪{y} is an independent set in G. We define the gated independence numbergi(G) of G to be the maximum cardinality of a gated independent set in G. We demonstrate that the gated independence number is closely related to both matching and domination parameters of graphs. We prove that the inequalities im(G)⩽gi(G)⩽mur(G) hold for every graph G, where im(G) and mur(G) denote the induced and uniquely restricted matching numbers of G. On the other hand, we show that γi(G)⩽gi(G) and γpr(G)⩽2gi(G) for every graph G without any isolated vertex, where γi(G) and γpr(G) denote the independence and paired domination numbers. Furthermore, we provide bounds on the gated independence number involving the order, size and maximum degree. In particular, we prove that gi(G)⩾n5 for every n-vertex subcubic graph G without any isolated vertex or any component isomorphic to K3,3, and gi(B)⩽3n8 for every n-vertex connected cubic bipartite graph B.

Further results on the hop domination number of a graph

A hop dominating set S in a connected graph G is called a minimal hop dominating set if no proper subset of S is a hop dominating set of G. The upper hop domination number γ+h (G) of G is the maximum cardinality of a minimal hop dominating set of G. Some general properties satisfied by this concept are studied. It is shown that for every two positive integers a and b where 2 ≤ a ≤ b, there exists a connected graph G such that γh(G) = a and γ+ h (G) = b. It is proved that minimal hop dominating set is NP-complete. It is proved that γh(G) and γ(G) are in general incomparable. It is shown that for every pair of positive integers a and b with a ≥ 2 and b ≥ 1, there exists a connected graph G such that γh(G) = a and γ(G) = b. We present an algorithm to compute minimal hop dominating set of G. Finally, we formulate an Integer linear programming problem to compute the hop domination number of G.

Domination on Bipolar Fuzzy Graph Operations: Principles, Proofs, and Examples

Bipolar fuzzy graphs, capable of capturing situations with both positive and negative memberships, have found diverse applications in various disciplines, including decision-making, computer science, and social network analysis. This study investigates the domain of domination and global domination numbers within bipolar fuzzy graphs, owing to their relevance in these aforementioned practical fields. In this study, we introduce certain operations on bipolar fuzzy graphs, such as intersection, join, and union of two graphs. Furthermore, we analyze the domination number and the global domination number for various operations on bipolar fuzzy graphs, including intersection, join, and union of fuzzy graphs and their complements.

A Feature-Selection Method Based on Graph Symmetry Structure in Complex Networks

This study aims to address the issue of redundancy and interference in data-collection systems by proposing a novel feature-selection method based on maximum information coefficient (MIC) and graph symmetry structure in complex-network theory. The method involves establishing a weighted feature network, identifying key features using dominance set and node strength, and employing the binary particle-swarm algorithm and LS-SVM algorithm for solving and validation. The model is implemented on the UNSW-NB15 and UCI datasets, demonstrating noteworthy results. In comparison to the prediction methods within the datasets, the model’s running speed is significantly reduced, decreasing from 29.8 s to 6.3 s. Furthermore, when benchmarked against state-of-the-art feature-selection algorithms, the model achieves an impressive average accuracy of 90.3%, with an average time consumption of 6.3 s. These outcomes highlight the model’s superiority in terms of both efficiency and accuracy.

حول بعد التقسيم وهيمنة الرسم البياني لعابد - وحيد 〖(AW)〗_r^4

يحتوي الرسم البياني H المرتبط ببساطة على مجموعة رؤوس بالنظر إلى الرسم البياني، فإن المجموعة المسيطرة S هي مجموعة فرعية من مجموعة الرؤوس V بحيث يكون أي رأس خارج S قريبًا من رأس واحد على الأقل داخل S. أصغر حجم من H هو المسيطر على المجموعات هو المعروف برقم سيطرة الرسم البياني. عندما يكون للرسم البياني H رأس x ومجموعة فرعية من الرأس V (H) ، يكون الفصل. فيما يتعلق بالقسم k المرتب من V (H) ، فإن الرسم التوضيحي لـ x بالنسبة إلى Π هو المتجهفي هذا البحث ، قمنا بدراسة بعض النتائج على رقم الهيمنة ورقم السيطرة المستقل والمقيد والمشار إليه بـ على التوالي في الرسوم البيانية Abid-Waheed والعلاقة بين رقم الهيمنة ورقم الهيمنة المستقلة والمقيدة. كما أن الهدف من هذه الورقة هو إنشاء أبعاد التقسيم لـ

Graphs obtained by disjoint unions and joins of cliques and stable sets

We consider the set of graphs that can be constructed from a one-vertex graph by repeatedly adding a clique or a stable set linked to all or none of the vertices added in previous steps. This class of graphs contains various well-studied graph families such as threshold, domishold, co-domishold and complete multipartite graphs, as well as graphs with linear clique-width at most 2. We show that it can be characterized by three forbidden induced subgraphs as well as by properties involving maximal stable sets and minimal dominating sets. We also give a simple recognition algorithm and formulas for the computation of the stability and domination numbers of these graphs.