Showing papers on "Dominating set published in 2023"

Improved local search for the minimum weight dominating set problem in massive graphs by using a deep optimization mechanism

Abstract: The minimum weight dominating set (MWDS) problem is an important generalization of the minimum dominating set problem with various applications. In this work, we develop an efficient local search scheme that can dynamically adjust the number of added and removed vertices according to the information of the candidate solution. Based on this scheme, we further develop three novel ideas to improve performance, resulting in our so-called DeepOpt-MWDS algorithm. First, we use a new construction method with five reduction rules to significantly reduce massive graphs and construct an initial solution efficiently. Second, an improved configuration checking strategy called CC 2 V3+ is designed to reduce the cycling phenomenon in local search. Third, a general perturbation framework called deep optimization mechanism (DeepOpt) is proposed to help the algorithm avoid local optima and to converge to a new solution quickly. Extensive experiments based on eight popular benchmarks of different scales are carried out to evaluate the proposed algorithm. Compared to seven state-of-the-art heuristic algorithms, DeepOpt-MWDS performs better on random and classic benchmarks and obtains the best solutions on almost all massive graphs. We investigate three main algorithmic ingredients to understand their impacts on the performance of the proposed algorithm. Moreover, we adapt the proposed general framework DeepOpt to another NP-hard problem to verify its generality and achieve good performance.

Complexity Results on Cosecure Domination in Graphs

Abstract: Let $$G=(V,E)$$ be a simple graph with no isolated vertices. A dominating set S of G is said to be a cosecure dominating set of G if for every vertex $$v \in S$$ there exists a vertex $$u \in V \setminus S$$ such that $$uv \in E$$ and $$(S \setminus \{v\}) \cup \{u\}$$ is a dominating set of G. The Minimum Cosecure Domination Problem is to find a minimum cardinality cosecure dominating set of G. Given a graph G and a positive integer k, the Cosecure Domination Decision Problem is to decide whether G has a cosecure dominating set of cardinality at most k. The Cosecure Domination Decision Problem is known to be NP-complete for bipartite, planar, and chordal graphs. In this paper, we show that the Cosecure Domination Decision Problem remains NP-complete for split graphs, an important subclass of chordal graphs. On the positive side, we present a linear-time algorithm to compute the cosecure domination number of cographs. In addition, we also study the approximation aspects of the Minimum Cosecure Domination Problem. We show that the problem can be approximated within an approximation ratio of $$(\varDelta +1)$$ for perfect graphs with maximum degree $$\varDelta $$ . We also prove that the problem cannot be approximated within an approximation ratio of $$(1-\epsilon )$$ ln(|V|) for any $$\epsilon >0$$ , unless P $$=$$ NP. Moreover, we prove that the Minimum Cosecure Domination Problem is APX-hard for bounded degree graphs.

Biomolecular and quantum algorithms for the dominating set problem in arbitrary networks

Abstract: A dominating set of a graph [Formula: see text] is a subset U of its vertices V, such that any vertex of G is either in U, or has a neighbor in U. The dominating-set problem is to find a minimum dominating set in G. Dominating sets are of critical importance for various types of networks/graphs, and find therefore potential applications in many fields. Particularly, in the area of communication, dominating sets are prominently used in the efficient organization of large-scale wireless ad hoc and sensor networks. However, the dominating set problem is also a hard optimization problem and thus currently is not efficiently solvable on classical computers. Here, we propose a biomolecular and a quantum algorithm for this problem, where the quantum algorithm provides a quadratic speedup over any classical algorithm. We show that the dominating set problem can be solved in [Formula: see text] queries by our proposed quantum algorithm, where n is the number of vertices in G. We also demonstrate that our quantum algorithm is the best known procedure to date for this problem. We confirm the correctness of our algorithm by executing it on IBM Quantum's qasm simulator and the Brooklyn superconducting quantum device. And lastly, we show that molecular solutions obtained from solving the dominating set problem are represented in terms of a unit vector in a finite-dimensional Hilbert space.

Input node placement restricting the longest control chain in controllability of complex networks

Abstract: The minimum number of inputs needed to control a network is frequently used to quantify its controllability. Control of linear dynamics through a minimum set of inputs, however, often has prohibitively large energy requirements and there is an inherent trade-off between minimizing the number of inputs and control energy. To better understand this trade-off, we study the problem of identifying a minimum set of input nodes such that controllabililty is ensured while restricting the length of the longest control chain. The longest control chain is the maximum distance from input nodes to any network node, and recent work found that reducing its length significantly reduces control energy. We map the longest control chain-constraint minimum input problem to finding a joint maximum matching and minimum dominating set. We show that this graph combinatorial problem is NP-complete, and we introduce and validate a heuristic approximation. Applying this algorithm to a collection of real and model networks, we investigate how network structure affects the minimum number of inputs, revealing, for example, that for many real networks reducing the longest control chain requires only few or no additional inputs, only the rearrangement of the input nodes.

Self-coalition graphs

Abstract: A coalition in a graph \(G = (V, E)\) consists of two disjoint sets \(V_1\) and \(V_2\) of vertices, such that neither \(V_1\) nor \(V_2\) is a dominating set, but the union \(V_1 \cup V_2\) is a dominating set of \(G\). A coalition partition in a graph \(G\) of order \(n = |V|\) is a vertex partition \(\pi = \{V_1, V_2, \ldots, V_k\}\) such that every set \(V_i\) either is a dominating set consisting of a single vertex of degree \(n-1\), or is not a dominating set but forms a coalition with another set \(V_j\) which is not a dominating set. Associated with every coalition partition \(\pi\) of a graph \(G\) is a graph called the coalition graph of \(G\) with respect to \(\pi\), denoted \(CG(G,\pi)\), the vertices of which correspond one-to-one with the sets \(V_1, V_2, \ldots, V_k\) of \(\pi\) and two vertices are adjacent in \(CG(G,\pi)\) if and only if their corresponding sets in \(\pi\) form a coalition. The singleton partition \(\pi_1\) of the vertex set of \(G\) is a partition of order \(|V|\), that is, each vertex of \(G\) is in a singleton set of the partition. A graph \(G\) is called a self-coalition graph if \(G\) is isomorphic to its coalition graph \(CG(G,\pi_1)\), where \(\pi_1\) is the singleton partition of \(G\). In this paper, we characterize self-coalition graphs.

Bipartite Domination Number of Mycielski Graph of Some Graph Families

Abstract: For a nontrivial connected graph G, a non-empty set S \(\subseteq\) V (G) is a bipartite dominating set of graph G, if the subgraph G[S] induced by S is bipartite and for every vertex not in S is dominated by any vertex in S. The bipartite domination number denoted by \(\gamma\)bip(G) of graph G is the minimum cardinality of a bipartite dominating set G. In this paper, we determine the exact bipartite domination number of a crown graph and its mycielski graph as well as the bipartite domination number of the mycielski graph of path and cycle graphs.

Total 2-domination number in digraphs and its dual parameter

Abstract: A subset $S$ of vertices of a digraph $D$ is a double dominating set (total $2$-dominating set) if every vertex not in $S$ is adjacent from at least two vertices in $S$, and every vertex in $S$ is adjacent from at least one vertex in $S$ (the subdigraph induced by $S$ has no isolated vertices). The double domination number (total $2$-domination number) of a digraph $D$ is the minimum cardinality of a double dominating set (total $2$-dominating set) in $D$. In this work, we investigate these concepts which can be considered as two extensions of double domination in graphs to digraphs, along with the concepts $2$-limited packing and total $2$-limited packing which have close relationships with the above-mentioned concepts.

Chemical Reaction Optimization for Minimum Weight Dominating Set

Abstract: Dominating set of a graph can be defined as the set of vertices that can cover all other vertices of the graph. The minimum weight dominating set (MWDS) is the minimum number of vertices in the dominating set with minimum total weight. In recent times, the chemical reaction optimization algorithm (CRO) has shown its supremacy in solving these types of problems. Therefore in this paper, a novel approach based on CRO has been proposed to solve the MWDS problem. The proposed method uses a repair-based technique to generate a molecule. To make the solution feasible by covering all vertices and to get better results, three supporting operators are implemented along with the CRO operators. Besides this, two repair operators are introduced. In the first repair operator, the searching procedure works based on the scaling properties of vertices, and the second one is a unique method for eliminating common neighbors of vertices of the dominating set. The performance of the proposed method is better than any other existing related algorithms. The performance is measured from different graphs of the benchmark datasets. It can be mentioned that the proposed method takes minimal running time to obtain the minimum weight compared to other benchmark methods.

Semitotal Domination Multisubdivision Number of a Graph

Abstract: A set S of vertices in G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number, $$\gamma _{t2}(G)$$ , is the minimum cardinality of a semitotal dominating set of G. The semitotal domination multisubdivision number of a graph G, $$msd_{\gamma _{t2}}(G)$$ , is the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the semitotal domination number of G. In this paper, we show that $$msd_{\gamma _{t2}}(G)\le 3$$ for any graph G of order at least 3, we also determine the semitotal domination multisubdivision number for some classes of graphs and characterize trees T with $$msd_{\gamma _{t2}}(T)=1$$ , 2 and 3, respectively.

Double total domination number of Cartesian product of paths

Abstract: A vertex set $ S $ of a graph $ G $ is called a double total dominating set if every vertex in $ G $ has at least two adjacent vertices in $ S $. The double total domination number $ \gamma_{\times 2, t}(G) $ of $ G $ is the minimum cardinality over all the double total dominating sets in $ G $. Let $ G \square H $ denote the Cartesian product of graphs $ G $ and $ H $. In this paper, the double total domination number of Cartesian product of paths is discussed. We determine the values of $ \gamma_{\times 2, t}(P_i\square P_n) $ for $ i = 2, 3 $, and give lower and upper bounds of $ \gamma_{\times 2, t}(P_i\square P_n) $ for $ i \geq 4 $.

Calculating the Centrality Values According to the Strengths of Entities Relative to their Neighbours and Designing a New Algorithm for the Solution of the Minimal Dominating Set Problem

Abstract: The dominating set problem in graph theory is an NP-complete problem for an arbitrary graph. There are many approximation-based studies in the literature to solve the dominating set problems for a given graph. Some of them are exact algorithms with exponential time complexities and some of them are based on approximation without robustness with respect to obtained solutions. In this study, the Malatya centrality value was used and a new Malatya centrality value was defined to solve the dominating set problem for a given graph. The improved algorithms have polynomial time and space complexities.

An Investigation of Corona Domination Number for Some Special Graphs and Jahangir Graph

Abstract: In this work, the study of corona domination in graphs is carried over which was initially proposed by G. Mahadevan et al. Let be a simple graph. A dominating set S 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 among all corona-dominating sets is called the corona-domination number and is denoted by (i.e) . In this work, the exact value of the corona domination number for some specific types of graphs are given. Also, some results on the corona domination number for some classes of graphs are obtained and the method used in this paper is a well-known number theory concept with some modification this method can also be applied to obtain the results on other domination parameters.

Some new generalizations of Domination using restrictions on degrees of vertices

Abstract: A set $D$ of vertices in a graph $G=(V,E)$ is a degree restricted dominating set for $G$ if each vertex $v_i$ in $D$ is dominating atmost $g(d_i)$ vertices of $V-D$, where $g$ is a function restricting the degree value $d_i$ with respect to the given function value $k_i$ for a natural valued function $f$ from the vertex set of the graph. We define three different types of Degree Restricted Domination by varying the way how the restricted function $g(v_i)$ is defined. If $g(d_i)=\big\lceil \frac{d_i}{k_i}\big\rceil$, the corresponding domination is called the ceil degree restricted domination, in short, $CDRD$, and the dominating set obtained in this manner is the $CDRD$-set. If $g(d_i)=\big\lfloor\frac{d_i}{k_i}\big\rfloor$ or $g(d_i)=d_i-k_i+1$, then the corresponding dominations are respectively called the floor degree restricted domination, in short $FDRD$, or the translate degree restricted domination, $TDRD$. The dominating sets obtained in this manner are the $FDRD$-set and the $TDRD$-set respectively. In this paper, we introduce these new generalizations of the domination number in line with the different $DRD$-sets and study these types of domination for some classes of graphs like complete graphs, caterpillar graphs etc. Degree restricted domination has a vital role in retaining the efficiency of nodes in a network and has many interesting applications.

Robust domination in random graphs

Abstract: In this paper, we study “robust” dominating sets of random graphs that retain the domination property even if a small deterministic set of edges are removed. We motivate our study by illustrating with examples from wireless networks in harsh environments. We then use the probabilistic method and martingale difference techniques to determine sufficient conditions for the asymptotic optimality of the robust domination number. We also discuss robust domination in sparse random graphs where the number of edges grows at most linearly in the number of vertices.

Counting independent dominating sets in linear polymers

Abstract: A vertex subset W⊆V of the graph G=(V,E) is an independent dominating set, if every vertex in V∖W is adjacent to at least one vertex in W and the vertices of W are pairwise non-adjacent. We enumerate independent dominating sets in several classes of graphs (polymer graph) made by a linear or cyclic concatenation of basic building blocks. Explicit recurrences are derived for the number of independent dominating sets of these kind of graphs. Generating functions for the number of independent dominating sets of triangular and squares cacti chain are also computed.

On the number of fair dominating sets of graphs

Abstract: Let G=(V,E) be a simple graph. A dominating set of G is a subset D⊆V such that every vertex not in D is adjacent to at least one vertex in D. The cardinality of a smallest dominating set of G, denoted by γ(G), is the domination number of G. For k≥1, a k-fair dominating set (kFD-set) in G, is a dominating set D such that |N(v)∩D|=k for every vertex v∈V∖D. A fair dominating set, in G is a kFD-set for some integer k≥1. In this paper, after presenting preliminaries, we count the number of fair dominating sets of some specific graphs.

Independent dominating sets in planar triangulations

Abstract: In 1996, Matheson and Tarjan proved that every planar triangulation on \(n\) vertices contains a dominating set %, i.e., a set \(S\) that contains a neighbor of every vertex not in \(S\), of size at most \(n/3\), and conjectured that this upper bound can be reduced to \(n/4\) when $n$ is sufficiently large. In this paper, we consider the analogous problem for independent dominating sets: What is the minimum \(\eps\) for which every planar triangulation on \(n\) vertices contains an independent dominating set of size at most \(\eps n\)? We prove that \(2/7 \leq \eps \leq 3/8\).

Generating Dominating Sets Using Locally-Defined Centrality Measures

Abstract: The dominating set problem has many practical applications but is well-known to be NP-hard. Therefore, there is a need for efficient approximation algorithms, especially in applications such as ad hoc wireless networks. Most distributed algorithms proposed in the literature assume that each node has knowledge of the network structure. We propose a distributed approximation algorithm that uses two rounds of communication, and where each node has only local information, both in terms of network structure and dominating set assignment. First, each node calculates a local centrality measure to determine whether it is part of the dominating set $D$. The second round guarantees $D$ is a dominating set by adding any non-dominated nodes. We compare several centrality measures and show that the Shapley value, introduced in game theory, is theoretically motivated and performs well in practice on several synthetic and real-world networks.

Further results on (total) restrained Italian domination

Abstract: An Italian dominating function on a graph [Formula: see text] is defined as a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text] or at least two vertices [Formula: see text] for which [Formula: see text]. The weight of an Italian dominating function is [Formula: see text]. The Italian domination number is the minimum weight taken over all Italian dominating functions of [Formula: see text] and denoted by [Formula: see text]. Three domination parameters related to the Italian dominating function are total Italian, restrained Italian, and total restrained Italian dominating function. A total ((restrained) (total restrained)) Italian dominating function [Formula: see text] is an Italian dominating function if the set of vertices with positive label ((the set of vertices with label [Formula: see text]), (at the same time, the set of vertices with positive label and the set of vertices with label [Formula: see text])) induces ((induces) (induce)) a subgraph with no isolated vertex. A minimum weight of any total ((restrained) (total restrained)) Italian dominating function [Formula: see text] is called a total ((restrained) (total restrained)) Italian domination number denoted by [Formula: see text], (([Formula: see text]) ([Formula: see text])). We initiate the study of parameters, restrained and total restrained Italian domination number of a graph [Formula: see text] and the middle graph of [Formula: see text]. For the family of standard graphs, we obtain the exact value of these parameters. For arbitrary graph [Formula: see text], we obtain the sharp bounds of these parameters, and for some corona graphs, we establish the precise value of these parameters.

A tight bound for independent domination of cubic graphs without 4‐cycles

Abstract: Given a graph G $G$ , a dominating set of G $G$ is a set S $S$ of vertices such that each vertex not in S $S$ has a neighbor in S $S$ . Let γ ( G ) $\gamma (G)$ denote the minimum size of a dominating set of G $G$ . The independent domination number of G $G$ , denoted i ( G ) $i(G)$ , is the minimum size of a dominating set of G $G$ that is also independent. We prove that if G $G$ is a cubic graph without 4-cycles, then i ( G ) ≤ 5 14 ∣ V ( G ) ∣ $i(G)\le \frac{5}{14}| V(G)| $ , and the bound is tight. This result improves upon two results from two papers by Abrishami and Henning. Our result also implies that every cubic graph G $G$ without 4-cycles satisfies i ( G ) γ ( G ) ≤ 5 4 $\frac{i(G)}{\gamma (G)}\le \frac{5}{4}$ , which supports a question asked by O and West.

Forcing 2-Metric Dimension in the Join and Corona of Graphs

Abstract: Let G be an undirected and connected graph with vertex set V(G). An ordered set of vertices {x1,...,xk} is a 2-resolving set in G if, for each distinct vertices u,v ∈ V(G), the lists of distances (dG(u,x1),..., dG(u,xk)) and (dG(v,x1),..., dG(v,xk)) differ in at least 2 positions. The minimum size of a 2-resolving set is the 2-metric dimension dim2(G) of G. A 2-resolving set of size dim2(G) is called a 2-metric basis for G. A subset S of a 2-metric basis W of G with the property that W is the unique 2-metric basis containing S is called a forcing subset of W. The forcing number fdim2(W) of W is the minimum cardinality of forcing subsets of W. The forcing number fdim2(G) of G is the smallest forcing number among all 2-metric basis of G. This study deals with the forcing subsets of 2-metric basis in graphs. The 2-metric basis in graphs resulting from some binary operations such as join and corona of graphs have been characterized. These characterizations are used to determine values for the forcing number of the 2-metric dimension of each graph considered.

Efficient Domination in Lattice graphs

Abstract: Given a graph $G$, a subset $S$ of vertices of $G$ is an efficient dominating set ($EDS$) if $|N[v] \cap S|=1,$ for all $v\in V(G)$. A graph $G$ is efficiently dominatable if it possesses an $EDS$. The efficient domination number of G is denoted by F(G) and is defined to be $\max \left\{\sum_{v \in S}(1 + \operatorname{deg} v):\right.$ $\left.S \subseteq V(G)\right.$ and $\left.|N[x] \cap S| \leq 1, \forall~ x \in V(G)\right\}$. In general, not every graph is efficiently dominatable. Further, the class of efficiently dominatable graphs has not been completely characterized and the problem of determining whether or not a graph is efficiently dominatable is NP-Complete. Hence, interest is shown to study the efficient domination property for graphs under restricted conditions or special classes of graphs. In this paper, we study the notion of efficient domination in some Lattice graphs, namely, rectangular grid graphs ($P_m \Box P_n$), triangular grid graphs, and hexagonal grid graphs.

Complete Cototal Roman Domination Number of a Graph for User Preference Identification in Social Media

Abstract: Many graph domination applications can be expanded to achieve complete cototal domination. If every node in a dominating set is regarded as a record server for a PC organization, then each PC affiliated with the organization has direct access to a document server. It is occasionally reasonable to believe that this gateway will remain available even if one of the scrape servers fails. Because every PC has direct access to at least two documents’ servers, a complete cototal dominating set provides the required adaptability to non-critical failure in such scenarios. In this paper, we presented a method for calculating a graph’s complete cototal roman domination number. We also examined the properties and determined the bounds for a graph’s complete cototal roman domination number, and its applications are presented. It has been observed that one’s interest fluctuate over time, therefore inferring them just from one’s own behaviour may be inconclusive. However, it may be able to deduce a user’s constant interest to some level if a user’s networking is also watched for similar or related actions. This research proposes a method that considers a user’s and his channel’s activity, as well as common tags, persons, and organizations from their social media posts in order to establish a solid foundation for the required conclusion.

An Approximation Algorithm for a Variant of Dominating Set Problem

Abstract: In this paper, we consider a variant of dominating set problem, i.e., the total dominating set problem. Given an undirected graph G=(V,E), a subset of vertices T⊆V is called a total dominating set if every vertex in V is adjacent to at least one vertex in T. Based on LP relaxation techniques, this paper gives a distributed approximation algorithm for the total dominating set problem in general graphs. The presented algorithm obtains a fractional total dominating set that is, at most, k(1+Δ1k)Δ1k times the size of the optimal solution to this problem, where k is a positive integer and Δ is the maximum degree of G. The running time of this algorithm is constant communication rounds under the assumption of a synchronous communication model.