Domination Number Research Articles

Convex Roman Dominating Functions on Graphs under some Binary Operations

Let $G$ be a connected graph. A function $f:V(G)\rightarrow \{0,1,2\}$ is a \textit{convex Roman dominating function} (or CvRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1 \cup V_2$ is convex. The weight of a convex Roman dominating function $f$, denoted by $\omega_{G}^{CvR}(f)$, is given by $\omega_{G}^{CvR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a CvRDF on $G$, denoted by $\gamma_{CvR}(G)$, is called the \textit{convex Roman domination number} of $G$. In this paper, we specifically study the concept of convex Roman domination in the corona and edge corona of graphs, complementary prism, lexicographicproduct, and Cartesian product of graphs.

Perfect Equitable Isolate Dominations in Graphs

A subset $S \subseteq V(G)$ is said to be a perfect equitable isolate dominating set of a graph $G$ if it is both perfect equitable dominating set of $G$ and isolate dominating set of $G$. The minimum cardinality of a perfect equitable isolate dominating set is called perfect equitable isolate domination number of $G$ and is denoted by $\gamma_{pe0}(G)$. A perfect equitable isolate dominating set $S$ of $G$ is called $\gamma_{pe0}$-set of $G$. In this paper, the authors give characterizations of a perfect equitable isolate dominating set of some graphs and graphs obtained from the join and corona of two graphs. Furthermore, the perfect equitable isolate domination numbers of these graphs is determined, and the graphs with no perfect equitable isolate dominating sets are investigated.

Formulas and Properties of 2-Hop Domination in Some Graphs

In this paper, 2-hop domination parameter is introduced and investigated on some special graphs and on the join of two graphs. Characterizations of 2-hop dominating sets in some special graphs are formulated to derive bounds or formulas of the parameter of these graphs. Moreover, new variant of pointwise non-domination is introduced to characterize 2-hop dominating sets in the join of two graphs. This characterization is used to calculate the exact value of 2-hop domination number of the join of two graphs.

On the number of minimum dominating sets and total dominating sets in forests

AbstractWe show that the maximum number of minimum dominating sets of a forest with domination number is at most and construct for each a tree with domination number that has more than minimum dominating sets. Furthermore, we disprove a conjecture about the number of minimum total dominating sets in forests by Henning, Mohr and Rautenbach.

A frequency and two-hop configuration checking-driven local search algorithm for the minimum weakly connected dominating set problem

A frequency and two-hop configuration checking-driven local search algorithm for the minimum weakly connected dominating set problem

Annihilator intersection graph of a lattice

For a lattice [Formula: see text], we associate a graph called the annihilator intersection graph of [Formula: see text], denoted by [Formula: see text] The vertex set of [Formula: see text] is the set of all nonzero zero-divisors of [Formula: see text] and any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text]. It has shown that the [Formula: see text] is disconnected if and only if the number of atoms in [Formula: see text] is two. If [Formula: see text] is connected, then we determine the diameter and the girth of [Formula: see text] We characterize all lattices whose annihilator intersection graph is planar. Further, we obtain the clique number and chromatic number of [Formula: see text] when [Formula: see text] is a finite Boolean lattice. We show that the domination number of [Formula: see text] is not exceeding two. Finally, we obtain a condition under which the annihilator intersection graph is identical with the zero-divisor graph and the annihilator ideal graph of lattices.

Approximation algorithm for the minimum partial connected Roman dominating set problem

Approximation algorithm for the minimum partial connected Roman dominating set problem

Topological Indices and Properties of the Prime Ideal Graph of a Commutative Ring and its Line Graph

Let R be a commutative ring with identity, and P be a prime ideal of R. The prime ideal graph, denoted by Γp, is the graph where the set of vertices is R/{0} and two vertices are joined by an edge if their product belongs to P. This paper will discuss topological indices and some properties of the prime ideal graph of a commutative ring and its line graph. Topological indices, such as the Wiener, first Zagreb, second Zagreb, Harary, Gutman, Schultz, and Harmonic indices, are related to the degree of vertices and the diameter of the graphs. In this paper, we also discuss the independence and domination numbers of the line graph.

Fractional Domination Number of the Fractal Graphs

Background/Objectives: A fractal is a collection of clearly defined structures that are uneven or irregular and can be divided into smaller pieces. Each of these smaller pieces has a relationship to the overall structure at random ranges that are analogous or identical. Fractal graphs are a highly effective way to construct well-defined patterns. The fractional domination number in a graph refers to the weight of a minimum fractional dominating function. The study is to find the bounds of the fractional domination and their related parameters in the fractal graphs. Methods: The sharp bounds and the constant ratio of fractional domination and their related parameters were calculated using an iterative process in all iterations of a self-similarity fractal graph. Findings: The study established the relation between the fractional domination number and other relevant parameters as fractional independence domination numbers for each iteration in the fractal graphs and explained how the concept of fractional domination works on fractal graphs such as the Koch snowflake and the Sierpinski triangle. Novelty: Previous researchers have studied bounds on maximum matching and dominating sets in fractal graphs such as the Koch snowflake and the Sierpinski triangle. The present study enhances the idea of finding the bounds of fractional domination parameters in a fractal graph. The construction of new beautiful ornaments, locating damaged cells for drug injection in medical science, placing stones in jewels, plumping work, and designing fractal antennas all can be made possible by this work in self-similarity fractal graphs. Keywords: Fractals, Fractional Dominating Functions, Fractional Domination Number, Fractional Independence Domination Number, Fractional Domination Chain

Fractional Star Domination Number of Graphs

Objectives: Consider a graph is the connected, undirected graph. This study creates a new parameter named the Fractional Star Domination Number (FRSDN), which is denoted by to calculate the minimum weight with the star of all vertices of and the function values of all edges. Methods: This study evaluates the of some standard graphs and bounds by generalizing the value that is provided by the function value of an edges. Findings: This study evaluated on some standard graphs, such as paths, cycles, and the rooted product of paths and cycles. Finally, we obtain some bounds on for some general graphs, as well as the exactness value yielded by them. For any graph without isolated vertices, we have . Novelty: The new parameter FRSDN of is created by combining the fractional dominating function and the star dominating set. Keywords: Domination Number, Star Domination Number, Fractional Dominating Function, Fractional Domination Number, Fractional Star Domination Number

Exploration of CCD-Number for Some Specific Graphs

Objectives: A dominating set S of a graph G is said to be a complementary corona dominating set (CCD-set) if every vertex in is either a pendent vertex or a support vertex. The smallest cardinality of a CCD-set is called the CCD-number and is denoted by . In this article, we explore the CCD-number for some specific graphs. Methods: We are finding the upper bound of and the lower bound of . We prove that both the upper and lower bound are equal for the given graphs. Findings: In this article, we obtained the CCD-number for some specific graphs. Novelty: Complementary corona dominating set is a vertex set such that the induced subgraph of the complementary set is isomorphic to corona virus. Keywords: Dominating Set, Corona Dominating Set, Support And Pendent Vertex, Butterfly Graph, Dumbbell Graph, Coconut Tree Graph, Peacock Head Graph, Sunlet Graph, Spider Graph, Firecracker Graph, Uniform g-ply graph

Generalization of CD-Number for Power Graph of Some Special Types of Tree Graphs

Objectives: The main objective of the article is to finding the corona domination for the power graph of some special types of tree graph. Method: A dominating set of a graph is said to be a corona dominating set if every vertex in is either a pendant vertex or a support vertex. The minimum cardinality of a corona dominating set is called the corona domination number and is denoted by . Findings: In this article, we study the -number for the power of PVB-tree and where and identify their exact values. Novelty: The corona domination was one of the recently developed domination parameter, along with this parameter we found the exact value for some special types of graphs. Keywords: Domination Number, Corona Domination Number, Total Domination, Power Graphs, Olive Tree And PVB­Tree

Upper g- Eccentric Domination in Fuzzy Graphs

Objectives: To introduce a contemporary aspect notorious as upper g-eccentric point set, upper g-eccentric number, upper g-eccentric dominating set, upper g-eccentric domination number in fuzzy graphs and related concepts. Methods: Strong arc in fuzzy graphs concepts are used to find upper g-eccentric number and upper g-eccentric domination number. Findings: The new idea upper g-eccentric domination in fuzzy graphs will leads to numerous application such as cell phone tower formation in remote areas. Novelty: Using the concepts g-distance, geodesics and strong arc in fuzzy graphs, the upper g-eccentric dominating set and upper g-eccentric domination number in fuzzy graphs are obtained. Some points are observed in this concept and some theorem related to the upper g-eccentric dominant set, its numbers are proved. Keywords: g-Eccentric Dominating Set; g-Eccentric Domination Number; Upper g-Eccentric Domination Number

Domination parameters and the removal of matchings

We consider the effect of the removal of a matching from a graph on the value of domination and domination-related parameters with emphasis on the removal of a maximum matching from a cubic graph or tree. For example, we show that the independent domination number of a nontrivial tree increases on the removal of any maximum matching.

Outer Triple Connected Corona Domination Number of Graphs

Background/ Objective: Given a graph G, a dominating set is said to be corona dominating set if every vertex such that or there exist a vertex if then . A corona dominating set is said to be an outer triple connected corona dominating set if any three vertices in lie on a path. The minimum cardinality taken over all the outer triple connected corona dominating sets of is called outer triple connected corona dominating number and it is denoted by . The study aims to find the outer triple connected corona domination number of some graphs. Method: To obtain outer triple connected corona domination number say m by proving and . To prove for a graph G we find a outer triple connected corona dominating set of G with cardinality m and then to prove we prove by contradiction. Findings: We investigated the above parameter for some derived graphs of path, cycle and wheel graph. Novelty : Outer triple connected corona domination number is a new concept in which the conditions of corona domination and triple connected are linked together. Keywords: Corona Domination, Pendent Vertex, Support Vertex, Triple Connected, Isolated Vertex, Pendant vertes

Mediated trust, the internet and artificial intelligence: Ideas, interests, institutions and futures

AbstractThis paper addresses the question of trust in communication, or mediated trust, with regard to the historical evolution of the Internet and, more recently, debates around the impacts of artificial intelligence (AI). At a conceptual level, it proposes a ‘Three I's’ framework of ideas, interests, and institutions as a way of understanding how and why current proposals for greater regulation of digital platforms counterpose questions around credibility and social licence for digital tech giants against a dominant set of ideas around the Internet as a privileged domain of free speech. By contrast, the rise of AI comes at a time when data‐driven business models associated with dominant platform businesses are in the ascendancy, so institutional arrangements need to be considered as a form of countervailing power to the capacity of tech giants to use AI to further consolidate forms of economic, political and communications power.

Inverse Fair Restrained Domination in the Join of Two Graphs

The inverse fair restrained domination number of G denoted by γ_frd^(-1) (G) is the minimum cardinality of an inverse fair restrained dominating set of G. An inverse fair restrained dominating set of cardinality γ_frd^(-1) (G) is called γ_frd^(-1)-set. In this paper, we investigate the concept and give some important results on inverse fair restrained dominating sets under the join of two graphs.

Modified Zagreb index and total domination in trees

Let [Formula: see text] be a simple connected graph. The modified first Zagreb index is denoted by [Formula: see text] is defined as [Formula: see text]. We present a lower bound for [Formula: see text] in terms of order and total domination number for trees and characterize the extremal trees attaining the bound.

Integer Linear Programming Models and Greedy Heuristic for the Minimum Weighted Independent Dominating Set Problem

This paper explores the Minimum Weighted Independent Dominating Set Problem and proposes novel approaches to tackle it. Namely, two integer linear programming formulations and a fast greedy heuristic as an alternative approach are proposed. Extensive computational experiments are conducted to evaluate the performance of these approaches on the established set of benchmark instances for the problem. The obtained results demonstrate that the introduced integer linear programming models are able to achieve optimal solutions on all instances with 100 nodes and significantly outperform existing exact methods on numerous other instances. Additionally, the greedy heuristic exhibits superior performance compared to competing greedy heuristics, particularly on random graphs. These findings suggest promising directions for future research, including the integration of these methods into hybrid algorithms or metaheuristic frameworks.

Secure Inverse Domination in the Corona and Lexicographic Product of Two Graphs

As secure domination and inverse domination garnered attention from various researchers, the combination of the two also raised a certain amount of curiosity. This paper aimed to investigate the secure inverse domination in graphs which is defined as follows. Let G be a connected simple graph and let D be a minimum dominating set of G. A dominating set S⊆V(G)∖D is an inverse dominating set of G with respect to D. The set S is called a secure inverse dominating set of G if for every u∈V(G)∖S, there exists v∈S such that uv∈E(G) and the set (S∖{v})∪{u} is a dominating set of G. The secure inverse domination number of G, denoted by γ_s^((-1) ) (G), is the minimum cardinality of a secure inverse dominating set of G. A secure inverse dominating set of cardinality γ_s^((-1) ) (G) is called γ_s^((-1) )-set. Particularly, the researchers examined and provided the characterization of secure inverse dominating set in the corona and lexicographic product of two graphs in this study. Moreover, the secure inverse domination number of graphs under the binary operations corona and lexicographic product were determined.