Domination Number Research Articles

Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables.

Chance constrained optimization problems allow to model problems where constraints involving stochastic components should only be violated with a small probability. Evolutionary algorithms have been applied to this scenario and shown to achieve high quality results. With this paper, we contribute to the theoretical understanding of evolutionary algorithms for chance constrained optimization. We study the scenario of stochastic components that are independent and normally distributed. Considering the simple single-objective (1+1) EA, we show that imposing an additional uniform constraint already leads to local optima for very restricted scenarios and an exponential optimization time. We therefore introduce a multi-objective formulation of the problem which trades off the expected cost and its variance. We show that multi-objective evolutionary algorithms are highly effective when using this formulation and obtain a set of solutions that contains an optimal solution for any possible confidence level imposed on the constraint. Furthermore, we prove that this approach can also be used to compute a set of optimal solutions for the chance constrained minimum spanning tree problem. In order to deal with potentially exponentially many trade-offs in the multi-objective formulation, we propose and analyze improved convex multi-objective approaches. Experimental investigations on instances of the NP-hard stochastic minimum weight dominating set problem confirm the benefit of the multi-objective and the improved convex multi-objective approach in practice.

Tree Domination Number of Semi Total Point Graphs

Tree Domination Number of Semi Total Point Graphs

Convex 2-Domination in Graphs

Let $G$ be a connected graph. A set $S \subseteq V(G)$ is \textit{convex $2$-dominating} if $S$ is both convex and $2$-dominating. The minimum cardinality among all convex $2$-dominating sets in $G$, denoted by $\gamma_{2con}(G)$, is called the \textit{convex $2$-domination number} of $G$. In this paper, we initiate the study of convex $2$- domination in graphs. We show that any two positive integers $a$ and $b$ with $6 \le a \le b$ are, respectively, realizable as the convex domination number and convex $2$-domination number of some connected graph. Furthermore, we characterize the convex $2$-dominating sets in the join, corona, lexicographic product, and Cartesian product of two graphs and determine the corresponding convex $2$-domination number of each of these graphs.

Differentiating Odd Dominating Sets in Graphs

Let $G=(V(G),E(G))$ be a simple and undirected graph. A dominating set \linebreak $S\subseteq V(G)$ is called a \textit{differentiating odd dominating set} if for every vertex $v\in V(G)$, \linebreak $|N[v]\cap S|\equiv 1(mod\ 2)$ and $N_G[u]\cap S\neq N_G[v]\cap S$ for every two distinct vertices $u$ and $v$ of $V(G)$. The minimum cardinality of a differentiating odd dominating set of $G$, denoted by $\gamma_D^o(G)$, is called the \textit{differentiating odd domination number}. In this paper, we discuss \linebreak differentiating odd dominating set in some graphs and give relationships between the \linebreak differentiating odd domination, odd domination, and differentiating-domination numbers. Moreover, we characterize the differentiating odd dominating sets in graphs resulting from join, corona, and lexicographic product of graphs and determine the differentiating odd domination numbers of these graphs.

Laplacian eigenvalue distribution, diameter and domination number of trees

For a graph G with domination number γ, Hedetniemi, Jacobs and Trevisan (2016) proved that m G [ 0 , 1 ) ≤ γ , where m G [ 0 , 1 ) means the number of Laplacian eigenvalues of G in the interval [ 0 , 1 ) . Let T be a tree with diameter d. In this paper, we show that m T [ 0 , 1 ) ≥ ( d + 1 ) / 3 . All trees achieving the lower bound are completely characterized. Moreover, we prove that the domination number of a tree is ( d + 1 ) / 3 if and only if it has exactly ( d + 1 ) / 3 Laplacian eigenvalues less than one. As an application, it also provides a new type of tree, which shows the sharpness of the inequality due to Hedetniemi, Jacobs and Trevisan.

On Minimal Geodetic Hop Domination in Graphs

Let $G$ be a nontrivial connected graph with vertex set $V(G)$. A set $S \subseteq V(G)$ is a geodetic hop dominating set of $G$ if the following two conditions hold for each $x\in V(G)\setminus S$: $(1)$ $x$ lies in some $u$-$v$ geodesic in $G$ with $u,v\in S$, and $(2)$ $x$ is of distance $2$ from a vertex in $S$. The minimum cardinality $\gamma_{hg}(G)$ of a geodetic hop dominating set of $G$ is the geodetic hop domination number of $G$. A geodetic hop dominating set $S$ is a minimal geodetic hop dominating set if $S$ does not contain a proper subset that is itself geodetic hop dominating set. The maximum cardinality of a minimal geodetic hop dominating set in $G$ is the \emph{upper geodetic hop domination number} of $G$, and is denoted by $\gamma^{+}_{hg}(G)$. This paper initiates the study of the minimal geodetic hop dominating set and the corresponding upper geodetic hop domination number of nontrivial connected graphs. Interestingly, every pair of positive integers $a$ and $b$ with $2\le a\le b$ is realizable as the geodetic domination number and the upper geodetic hop domination number, respectively, of some graph. Furthermore, this paper investigates the concept in the join, corona and lexicographic product of graphs.

On the <i>k</i>-restricted Intersection Graph

The problem on intersection graph was introduced by Szpilrajn-Marczewski in 1945. This study introduces a new variant of intersection graph, called the $k$-restricted intersection graph. Let $\mathcal{S}_n$ be a nonempty $n$-element set, for some positive integer $n$ and let $S_{(n,k)}$ be the set of all the $k$-element subsets of $\mathcal{S}_n$ where $0\leq k\leq n$. A $k$-restricted intersection graph, denoted by $G_{S_{(n,k)}}$, is a graph with vertex set $S_{(n,k)}$ such that two vertices $A,B\in S_{(n,k)}$ are adjacent whenever $A\cap B\neq \emptyset$ and $A\neq B$. Here, we determined the order and size of $G_{S_{(n,k)}}$. Moreover, some parameters such as independence number, domination number, and isolate domination number of the $k$-restricted intersection graph were established. Finally, necessary and sufficient conditions for a $G_{S_{(n,k)}}$ to be isomorphic to the cycle graph and complete graph were determined.

Study on Even Sum Domination Number of Some Graphs

Let G be a connected graph and uv ∈ E(G). We say, the vertex v even sum dominates u (u even sum dominates v) if deg(v)+deg(u) is an even number. A set S is an even sum dominating set (ESDS) if every vertex v ∈ V is either in S or even sum dominated by a vertex in S. An even sum dominating set S is a mimimal even sum dominating set if no proper subset S′⊂ S is an even sum dominating set. The even sum domination number γes(G) of a graph G is the minimum cardinality of an even sum dominating set of G. In this paper, we discuss some properties and bounds for this concept. We also derive even sum domination number for some standard graphs.

Closed Geodetic Hop Domination in Graphs

Let G be a simple, undirected and connected graph. A subset S ⊆ V (G) is a geodetic cover of G if IG[S] = V (G), where IG[S] is the set of all vertices of G lying on any geodesic between two vertices in S. A geodetic cover S of G is a closed geodetic cover if the vertices in S are sequentially selected as follows: Select a vertex v1 and let S1 = {v1}. If G is nontrivial, select a vertex v2 ̸= v1 and let S2 = {v1, v2}. Where possible, for i ≥ 3, successively select vertex vi ∈/ IG[Si−1] and let Si = {v1, v2, ..., vi}. Then there exists a positive integer k such that Sk = S. A geodetic cover S of G is a geodetic hop dominating set if every vertex in V (G) \ S is of distance2 from a vertex in S. A geodetic hop dominating set S is a closed geodetic hop dominating set if S is a closed geodetic cover of G. The minimum cardinality of a (closed) geodetic hop dominating set of G is the (closed) geodetic hop domination number of G. This study initiates the study of the closed geodetic hop domination. First, it characterizes all graphs G of order n whose closed geodetic hop domination numbers are 2 or n, and determines the closed geodetic hop domination number of paths, cycles and multigraphs. Next, it shows that any positive integers a and b with 2 ≤ a ≤ b are realizable as the closed geodetic number and closed geodetic hop domination number of a connected graph. Also, every positive integer n, m and k with 4 ≤ m ≤ k and 2k−m+2 ≤ n are realizable as the order, geodetic hop domination number and closed geodetic hop domination number, respectively of a connected graph. Furthermore, the study characterizes the closed geodetic hop dominating sets of graphs resulting from the join, corona and edge corona of graphs.

Some Properties of Dominant Local Metric Dimension

Let $G$ be a connected graph with vertex set $V$. Let $W_l$ be an ordered subset defined by $W_l=\{w_1,w_2,\dots,w_n\}\subseteq V(G)$. Then $W_l$ is said to be a dominant local resolving set of $G$ if $W_l$ is a local resolving set as well as a dominating set of $G$. A dominant local resolving set of $G$ with minimum cardinality is called the dominant local basis of $G$. The cardinality of the dominant local basis of $G$ is called the dominant local metric dimension of $G$ and is denoted by $Ddim_l(G)$. We characterize the dominant local metric dimension for any graph $G$ and for some commonly known graphs in terms of their domination number to get some properties of dominant local metric dimension.

Finding the domination number of triangular belt networks

Domination in graphs has been an extensively researched branch of graph theory. A set S⊆V is said to be the dominating set of graph G if for every v∈V-S there exists a vertex u∈S such that uv∈E. The minimum cardinality of vertices among the dominating set of G is called the domination number of G denoted by γ(G). The main object of this paper is to study the domination number of some networks such as triangular belt networks TBn, alternate triangular belt networks ATBn and restricted square of bistar networks Bn,n.

Employer Dominance and Worker Earnings in Finance.

A few large firms in the U.S. financial system achieve substantial economic gains. Their dominance sets them apart while also raising concerns about the suppression of worker earnings. Utilizing administrative data, this study reveals that the largest financial firms pay workers an average of 30.2% more than their smallest counterparts, significantly exceeding the 7.9% disparity in nonfinance sectors. This positive size-earnings relationship is consistently more pronounced in finance, even during the 2008 crisis or compared to the high-tech sector. Evidence suggests that large financial firms' excessive gains, coupled with their workers' sought-after skills, explain this distinct relationship. (JEL G20, J31, J42, L11, L12, L13).

A Local Search Algorithm with Vertex Weighting Strategy and Two-Level Configuration Checking for the Minimum Connected Dominating Set Problem.

The minimum connected dominating set problem is a combinatorial optimization problem with a wide range of applications in many fields. We propose an efficient local search algorithm to solve this problem. In this work, first, we adopt a new initial solution construction method based on three simplification rules. This method can reduce the size of the original graph and thus obtain a high-quality initial solution. Second, we propose an approach based on a two-level configuration checking strategy and a tabu strategy to reduce the cycling problem. Third, we introduce a perturbation strategy and a vertex weighting strategy to help the algorithm be able to jump out of the local optimum effectively. Fourth, we combine the scoring functions Cscore and Mscore with the aforementioned strategies to propose effective methods for selecting vertices. These methods assist the algorithm in selecting vertices that are suitable for addition to or removal from the current candidate solution. Finally, we verify the performance advantages of the local search algorithm by comparing it with existing optimal heuristic algorithms on two sets of instances. The experimental results show that the algorithm exhibits better performance on two sets of classical instances.

Domination number of Cartesian product through space projections

Domination number of Cartesian product through space projections

Metric dimension of unit graphs

The notion of the metric dimension of a graph is well-known and its study is well entrenched in the literature. In this paper, we introduce new classes of graphs that exhibit remarkable characteristic: the metric dimension and the diameter of the unit graph are equal. Additionally, we provide a characterization of finite commutative rings [Formula: see text], wherein the metric dimension assumes a value [Formula: see text], where [Formula: see text]. Further, we also demonstrate an exhaustive examination that ascertains the precise finite commutative rings [Formula: see text], in which domination number is equal to metric dimension of unit graph.

Variants of the domination number for flower snarks

Variants of the domination number for flower snarks

Some Bench Mark Results on Total Domination Subdivision Stable Graph

For a graph G, the total dominating set defined as a set of vertices in S such that all the vertices in V(G) has at least one neighbor in S, the least cardinality is noted as t(G). The total domination number of each and every graph while subdividing any edge xy of G is equal to the total domination number of G, which results in the total domination subdivision stable graph abbreviated as TDSS and the symbolic expression is Gtsd(xy). The research paper, we introduce TDSS and proposed conditions under which a graph is TDSS and not TDSS.

Eternal 1-security number of a fuzzy graph with level J

The eternal 1-secure set, strategically protects graphical structures against attacks by placing sentinels within dominating sets. This study adapts this concept to real-world scenarios, like emergency response systems, by addressing challenges such as roadblocks and traffic congestion. We introduce the secure set and the eternal 1-secure set for fuzzy graphs based on edge membership levels J. Our analysis examines the relationship between the security number of a fuzzy graph G at level J and the domination number of its spanning subgraph at level J, demonstrating the potential of the eternal 1-secure set to enhance security measures in dynamic environments.

Some novel concepts of intuitionistic fuzzy directed graphs with application in selecting a suitable place for opening restaurant

In this manuscript, we first initiate several types of effective arcs of intuitionistic fuzzy directed graphs, followed by discussions on different types of dominations in intuitionistic fuzzy directed graphs and their application in decision-making. The notion of dominations in fuzzy graphs, fuzzy directed graphs, intuitionistic fuzzy graphs and picture fuzzy graphs have been extensively discussed in the literature. Thus, the work presented in our study is two-fold: on one side, it extends the notion of domination in fuzzy directed graphs, while on the other side, it fills the gap existing in the literature. Initially, we introduce several types of effective arcs like semi-λ effective arc, semi-ν effective arcs in intuitionistic fuzzy directed graphs. Subsequently, we initiate the notions of dominations and domination numbers for several types of intuitionistic fuzzy directed graphs based on these effective arcs. We present numerous significant characterizations of dominations in intuitionistic fuzzy directed graphs by utilizing minimal dominating sets. Furthermore, we investigate the minimal dominating sets and domination numbers of intuitionistic fuzzy-dipaths and dicycles, and provide various fascinating results. Finally, by utilizing the concepts presented in this study, we provide the algorithm for the solution of the problem related to decision-making regarding the opening of restaurants in various areas of the different cities.

On maximal Roman domination in graphs: complexity and algorithms

For a simple undirected connected graph G = (V, E), a maximal Roman dominating function (MRDF) of G is a function f : V (G) → {0, 1, 2} with the following properties: (i) For every vertex v ∈ {v ∈ V|f(v) = 0}, there exists a vertex u ∈ N(v) such that f(u) = 2. (ii) The set {v ∈ V|f(v) = 0} is not a dominating set of G; In other words, there exists a vertex v ∈ {v ∈ V|f(v) ≠ 0} such that N(v) ∩ {u ∈ V|f(u) = 0} ∅. The weight of an MRDF of G is the sum of its function values over all vertices, denoted as f(G) = ∑v∈V (G) f(v), and the maximal Roman domination number of G, denoted by γmR(G), is the minimum weight of an MRDF of G. In this paper, we establish some bounds of the maximal Roman domination number of graphs. Additionally, we develop an integer linear programming formulation to compute the maximal Roman domination number of any graph. Furthermore, we prove that maximal Roman domination problem (MRD) is NP-complete even restricted to star convex bipartite graphs and chordal bipartite graphs. Lastly, we show the maximal Roman domination number of threshold graphs, trees, and block graphs can be computed in linear time.