Domination Number Research Articles

More results on the outer-independent triple Roman domination number

More results on the outer-independent triple Roman domination number

Further results on outer independent triple Roman domination

An outer-independent triple Roman dominating function (OI[3]RDF) on a graph G = ( V , E ) is function f : V → { 0 , 1 , 2 , 3 , 4 } having the property that (i) if f ( v ) = 0 , then v must have either a neighbor assigned 4 or two neighbors one of which is assigned 3 and the other at least 2 or v has three neighbors all assigned 2; (ii) no two vertices assigned 0 are adjacent; (iii) if f ( v ) = 1 , then v must have either a neighbor assigned at least 3 or two neighbors assigned 2; (iv) if f ( v ) = 2 , then v must have one neighbor assigned at least 2. The weight of an OI[3]RDF is the sum of its function value over the whole set of vertices, and the outer-independent triple Roman domination number of G is the minimum weight of an OI[3]RDF on G. In this paper, we continue the study of outer-independent triple Roman domination number in graphs. First, we characterize all graphs with small or large outer-independent triple Roman domination number, and then we establish relationships with some other related parameters. Finally we present a sharp upper bound for the outer-independent triple Roman domination number of trees.

On the annihilator graphs of partial transformation semigroups

Let X n = { 1 , 2 , … , n } and P n be a partial transformation semigroup on X n . Obviously, the empty set ∅ is a zero element of P n and denoted by 0. Let P n * = P n \\ { 0 } . An element α ∈ P n is a zero divisor of P n if there exists β ∈ P n * such that α β = 0 = β α . The set Ann ( α ) = { β ∈ P n : α β = 0 = β α } is called the annihilator of α in P n . It is clear that 0 ∈ Ann ( α ) for all α ∈ P n . Let Z ( P n ) be the set of all zero divisors of P n and Z ( P n ) * = Z ( P n ) \\ { 0 } . Generally, if α ∈ Z ( P n ) , then there exists β ∈ Z ( P n ) * in which β ∈ Ann ( α ) . In this paper, we construct the annihilator graph Γ n of P n which is an undirected simple graph with vertex set Z ( P n ) * and two distinct vertices α and β are adjacent if and only if Ann ( α ) ∩ Ann ( β ) ≠ { 0 } . Furthermore, we prove some basic structural properties of Γ n and determine invariants for Γ n such as the diameter, girth, clique, domination number, and independence number.

A Comparative Study: Domination Parameters and Physico-Chemical Properties of Benzenoid Hydrocarbons

In graph theory, a graph is a mathematical structure that consists of a set of vertices and a set of edges connecting those pairs of vertices. Let G = ( V , E ) be a graph, a dominating set for G is a subset D ⊆ V of vertices such that every vertex in V ∖ D has at least one neighbor in D. The domination number of G, denoted by γ ( G ) , is the minimum cardinality of a dominating set in G. Recently, Khan (2023) performed a comparative study of seven important domination parameters with the π -electronic energy of benzenoid hydrocarbons, with a new theoretical approach. Physico-chemical properties are crucial for understanding the physical and chemical behavior of a molecule. In this paper, we perform a comparative study of those seven domination parameters with the physico-chemical properties of benzenoid hydrocarbons. We find the values of those domination parameters for the 22 lower benzenoid hydrocarbons and follow the statistical approach described by Khan (2023). The two physico-chemical properties to be chosen are the normal boiling point T b , a representative for intermolecular and van-der-Waals type interactions, whereas, the standard enthalpy/heat of formation Δ H f o is referred to represent the thermal properties of a molecular graph. This study further enhances that the performance of paired domination number is exceptional among all other parameters and highly correlated with both of the physico-chemical properties. Its correlation coefficient is 0.9981 (respectively, 0.9726) with T b (respectively, Δ H f o ). The outstanding correlation coefficient of paired domination number ensures further usage in quantitative structure-property relationship (QSPR) models.

Shock petrographic and numerical modeling constraints on the morphology and size of the Morokweng impact structure, South Africa

AbstractThe 369 m deep M4 drill hole, located ~18 km NNW of the center of the 146 Ma Morokweng impact structure (MIS), intersects shocked Archean granitoid gneisses and subsidiary dolerite intrusions that are cut by faults, cataclasites and mm‐ to m‐wide suevitic and pseudotachylitic breccia dikes. The shock features in quartz in the gneisses and breccia dikes include decorated planar deformation features (PDFs), planar fractures, feather features, and toasting. Other minerals show features that may be shock‐related, such as multiple sets of planar features and alternate twin ladder structures in feldspars, kink bands in biotite, and planar features in titanite, apatite, and zircon; however, these are variably annealed and/or overprinted by hydrothermal alteration effects, and confirmation of their origin awaits further study. Universal Stage measurements of PDF sets in quartz from 12 gneissic target rocks and from lithic and mineral clasts in three suevitic and three pseudotachylitic breccia dikes reveal four dominant sets: (0001), {}, {} and {}. Based on these observations, the average peak shock pressure in these rocks is estimated at ≤16 GPa, which supports the original proximity (within one transient cavity radius) of these rocks to the point of impact. No discernible depth‐dependent shock attenuation was noted in the core. These shock levels and the elevated structural position of the rocks in the M4 core relative to the impact melt sheet intersected in drill holes closer to the center of the MIS suggest that the M4 lithologies represent part of the parautochthonous peak ring volume that subsequently experienced 1.5–2 km of post‐impact erosion before it was buried beneath younger sediments. Numerical modeling using the iSALE‐2D code suggests that the original Morokweng crater had a rim‐to‐rim diameter of between 70 and 80 km, and that the rocks in the M4 core were originally located at a depth of 7–8 km and a radial distance of 8–9 km from the point of impact.

An approximation algorithm for the prize-collecting connected dominating set problem

An approximation algorithm for the prize-collecting connected dominating set problem

Dominion on Grids

The domination (number) of a graph G=(V,E), denoted by γ(G), is the size of the minimum dominating sets of V(G), also known as γ-sets. As such, the dominion of G, denoted by ζ(G), counts all its γ-sets. We proved a conjecture from one of the authors on the dominion of cycles C3k−1 and C3k−2, k≥2. Further, we found the formulae and recurrence relations for the dominions of several grids, Gm,n, with 2≤m≤4 and other results when m≤9 and n≤20. In general, domination and dominion play important roles in assessing certain vulnerabilities of any given network system.

Connected Co-Independent Hop Domination in the Edge Corona and Complementary Prism of Graphs

Let $G$ be a connected graph. A subset $S$ of $V(G)$ is a connected co-independent hop dominating set in $G$ if the subgraph induced by $S$ is connected and $V(G) \backslash S$ is an independent set where for each $v \in V(G) \backslash S$, there exists a vertex $u \in S$ such that $d_G(u,v)=2$. The smallest cardinality of such an $S$ is called the connected co-independent hop domination number of $G$. This paper presents the characterizations of the connected co-independent hop dominating sets in the edge corona and complementary prism of graphs and determines the exact values of their corresponding connected co-independent hop domination number.

On greedy approximation algorithm for the minimum resolving dominating set problem

On greedy approximation algorithm for the minimum resolving dominating set problem

Enhancing Network Security in Distributed Systems Using Middle Roman Dominating Functions

A Middle Roman dominating function (MRDN) on a graph G = (V,E) is a function f:v→{0,1,2,3} satisfying the condition that every vertex u with f(u)=0 is adjacent to at most one vertex v with f(v)=2 or 3. Further if a vertex is assigned 2, then at most two of its vertices can be assigned 0 and if a vertex is assigned 3, then all its neighbours can be assigned 0. The weight of a MRDF is the value f(V(G))=∑_uϵV▒〖f(u〗). The Middle Roman domination number γ_MR (G) is the minimum weight of a MRDF on G. In this paper, we introduce Middle Roman Domination number denoted as γ_MR (G), study the properties of the function, present some characterization and determine γ_MR (G)-value for some graphs. Middle Roman Dominating Functions (MRDFs) present a unique approach within graph theory, with significant implications for various fields in computer science. By assigning values to vertices under specific constraints, MRDFs enable the optimization of resources, enhancing network security, load balancing, and energy efficiency in distributed systems. This paper explores the application of MRDFs in scenarios such as intrusion detection, resilient network design, task scheduling, and sensor activation. By minimizing the overall weight while maintaining functional requirements, MRDFs provide an effective strategy for addressing challenges in network topology, resource allocation, and fault tolerance. The versatility of MRDFs makes them a valuable tool in the development of robust and efficient computing systems.

Some results on the super domination number of a graph II

Some results on the super domination number of a graph II

Connected power domination number of product graphs

Connected power domination number of product graphs

On double domination numbers of signed complete graphs

Let [Formula: see text] be a signed graph with a vertex set [Formula: see text]. A set [Formula: see text] is said to be a double dominating set of [Formula: see text] if it satisfies the following conditions: (i) [Formula: see text] for each [Formula: see text], and (ii) [Formula: see text] is balanced, where [Formula: see text] denotes the closed neighborhood of [Formula: see text] and [Formula: see text] denotes the subgraph induced by the edges of [Formula: see text] with one end vertex in [Formula: see text] and the other end vertex in [Formula: see text]. The minimum size among all the double dominating sets of [Formula: see text] is the double domination number [Formula: see text] of [Formula: see text]. In this study, we investigated this parameter for signed complete graphs. We prove that, for [Formula: see text], if [Formula: see text] is a signed complete graph, then [Formula: see text] and these bounds are sharp. Moreover, for all signed complete graphs over [Formula: see text] we determined their possible double domination numbers. Finally, we compute the double domination numbers of all signed complete graphs of orders up to six.

Maker-Breaker domination game played on corona products of graphs

In the Maker-Breaker domination game, Dominator and Staller play on a graph G by taking turns in which each player selects a not yet played vertex of G. Dominator’s goal is to select all the vertices in a dominating set, while Staller aims to prevent this from happening. In this paper, the game is investigated on corona products of graphs. Its outcome is determined as a function of the outcome of the game on the second factor. Staller-Maker-Breaker domination numbers are determined for arbitrary corona products, while Maker-Breaker domination numbers of corona products are bounded from both sides. All the bounds presented are demonstrated to be sharp. Corona products as well as general graphs with small (Staller-)Maker-Breaker domination numbers are described.

The 2-rainbow domination number of Cartesian product of cycles

The 2-rainbow domination number of Cartesian product of cycles

A distributed algorithm for secure dominating set problem in IoT networks

A distributed algorithm for secure dominating set problem in IoT networks

On the strong domination number of proper enhanced power graphs of finite groups

On the strong domination number of proper enhanced power graphs of finite groups

MDSA: A Dynamic and Greedy Approach to Solve the Minimum Dominating Set Problem

The graph theory is one of the fundamental structures in computer science used to model various scientific and engineering problems. Many problems within the graph theory are categorized as NP-hard and NP-complete. One such problem is the minimum dominating set (MDS) problem, which seeks to identify the minimum possible subsets in a graph such that every other node in the subset is directly connected to a node in this subset. Due to its inherent complexity, developing an efficient polynomial-time method to address the MDS problem remains a significant challenge in graph theory. This paper introduces a novel algorithm that utilizes a centrality measure known as the Malatya Centrality to effectively address the MDS problem. The proposed algorithm, called the Malatya Dominating Set Algorithm (MDSA), leverages centrality values to identify dominating sets within a graph. It extends the Malatya centrality by incorporating a second-level centrality measure, which enhances the identification of dominating nodes. Through a systematic and algorithmic approach, these centrality values are employed to pinpoint the elements of the dominating set. The MDSA uniquely integrates greedy and dynamic programming strategies. At each step, the algorithm selects the most optimal (or near-optimal) node based on the centrality values (greedy approach) while updating the neighboring nodes’ criteria to influence subsequent decisions (dynamic programming). The proposed algorithm demonstrates efficient performance, particularly in large-scale graphs, with time and space requirements scaling proportionally with the size of the graph and its average degree. Experimental results indicate that our algorithm outperforms existing methods, especially in terms of time complexity when applied to large datasets, showcasing its effectiveness in addressing the MDS problem.

A Greedy Simulated Annealing-based Multiobjective Algorithm for the Minimum Weight Minimum Connected Dominating Set Problem

A Greedy Simulated Annealing-based Multiobjective Algorithm for the Minimum Weight Minimum Connected Dominating Set Problem

1-MOVABLE 2-OUTER-INDEPENDENT DOMINATING SETS IN GRAPHS

A set is a 1-movable 2-outer-independent dominating set of if is a 2-outer-independent dominating set of and for every is a 2-outer-independent dominating set of or there exists a vertex such that is a 2 -outer-independent dominating set of . The 1-movable 2 -outerindependent domination number of a graph , denoted by , is the smallest cardinality of a 1-movable 2-outer-independent dominating set of . A 1 -movable 2 -outer-independent dominating set of with cardinality equal to is called -set of . We characterize 1-movable 2 -outer-independent dominating set in a graph and the 1-movable 2-outer-independent domination in some special graphs.