Power Dominating Set Research Articles

Power domination in generalized Mycielskian of spiders

The power domination problem in the graph was introduced to model the monitoring problem in electric networks. It is to find a set of vertices with minimum cardinality, called a power dominating set (PDS), that monitors the whole set $ V $ after applying the rules, domination and propagation. In this paper, we discuss the power domination problem in generalized Mycielskian of spiders. We characterize spiders with $\gamma _P(\mu _m(T))= 1$. If $\gamma _P(\mu _m(T))\neq 1 $ and $ m $ is even then $ \gamma _P(\mu _m(T))=\frac {m}{2} +1$. We also show that if $ m $ is odd and $ \gamma _P(\mu _m(T))\neq 1 $, then $ \lceil \frac {m+1}{2}\rceil \leq \gamma _P(\mu _m(T))\leq \lceil \frac {m}{2}\rceil +1$.

2-power domination number for Knödel graphs and its application in communication networks

In a graph G, if each node v ∈ V(G) \ S is connected to some node in S, then the set S of nodes is referred to as a dominating set. The domination number of G is the minimum cardinality of all dominating sets of G and is represented by γ(G). If a dominating set S monitors every node in the system under a set of guidelines for power systems monitoring, then the set S is referred to as a power-dominating set of G. The power domination number of G is the least number of vertices of a power dominating set of G. A generalization of power domination is the k-power domination in a graph G. The k-power domination number of G is the minimum cardinality of all k-power dominating sets of G and is represented by γp,k(G). In this paper, we have obtained the 2-power domination number represented by γp,2(G) for 4-regular Knödel graphs and given the lower bound for 5-regular Knödel graphs.

Resolving-power domination number of probabilistic neural networks

Power utilities must track their power networks to respond to changing demand and availability conditions to ensure effective and efficient operation. As a result, several power companies continuously employ phase measuring units (PMUs) to continuously check their power networks. Supervising an electric power system with the fewest possible measurement equipment is precisely the vertex covering graph-theoretic problems otherwise a variation of the dominating set problem, in which a set D is defined as a power dominating set (PDS) of a graph if it supervises every vertex and edge in the system with a couple of rules. If the distance vector eccentrically characterizes each node in G with respect to the nodes in R, then the subset R of V (G) is a resolving set of G. The problem of finding power dominating set and resolving set problems are proven to be NP-complete in general. The finite subset R of V (G) is said to be resolving-power dominating set (RPDS) if it is both resolving and power dominating set, which is another NP-complete problem. The ηp (G) is the minimal cardinality of an RPDS of a graph G. A neural network is a collection of algorithms that tries to figure out the underlying correlations in a set of data by employing a method that replicates how the human brain functions. Various neural networks have seen rapid progress in multiple fields of study during the last few decades, including neurochemistry, artificial intelligence, automatic control, and informational sciences. Probabilistic neural networks (PNNs) offer a scalable alternative to traditional back-propagation neural networks in classification and pattern recognition applications. They do not necessitate the massive forward and backward calculations that ordinary neural networks entail. This paper investigates the resolving-power domination number of probabilistic neural networks.

Optimal PMU placement problem in octahedral networks

Power utilities must track their power networks to respond to changing demand and availability conditions to ensure effective and efficient operation. As a result, several power companies employ phase measuring units (PMUs) to check their power networks continuously. Supervising an electric power system with the fewest possible measurement equipment is precisely the vertex covering graph-theoretic problem, in which a set D is defined as a power dominating set (PDS) of a graph if it supervises every components (vertices and edges) in the system (with a couple of rules). The γp(G) is the minimal cardinality of a PDS of a graph G. In this present study, the PDS is identified for octahedral networks.

Power domination in Mycielskian of spiders

The power domination problem in graphs consists of finding a minimum set of vertices that monitors the entire graph G governed by two ‘monitoring rules’- domination and propagation. A set is a power dominating set (PDS) if it can monitor all vertices of G. The minimum cardinality of a PDS of G is called the power domination number, , of G. In this paper, we study the power domination problem in Mycielskian of spiders. For a spider T, we have and . We characterize spiders, T, for which and

Robustness Analysis Of Structural Controllability for Directed Networks Against Single Edge Attacks

Infrastructure systems are an essential component, evolving with greater interconnectivity and interdependence at varying degrees. The control robustness of a network against malicious attack and random failure also becomes a further considerable problem in network controllability and its robustness. An adversary who is adequately knowledgeable about the control system can take control of aspects of the network as it can compromise the control network’s subset of critical nodes and/or disconnect parts of the control network resulting in low observability. Therefore, safeguarding critical infrastructure systems from different disruptions is primarily significant. This paper focuses the POWER DOMINATING SET (PDS) problem, originally introduced by Haynes to study the structure of electric power network control systems and their efficient control, as an alternate framework for the examination of the structural controllability of networks. However, PDS is generally known to be NP-complete with low approximability with recent work focusing on studying properties of restricted graph classes. Based on the PDS problem, this paper also is dedicated to studying the different edge attack strategies, as well as the robustness of network controllability of Erd s- e n i networks with directed control links under single edge attacks. MATLAB will be utilised in order to produce a simulative evaluation for more realistic critical infrastructure networks such as real power networks.

Equitable power domination number of total graph of certain graphs

Let G (V, E), or simply G, be a graph. A set S ⊆ V is said to be a power dominating set (PDS) if every vertex u ∈ V − S is observed by certain vertices in S by the following two rules: (a) if a vertex v in G is in PDS, then it dominates itself and all the adjacent vertices of v and (b) if an observed vertex v in G has k > 1 adjacent vertices and if k − 1 of these vertices are already observed, then the remaining one non-observed vertex is also observed by v in G. A power dominating set S ⊆ V in G is said to be an equitable power dominating set (EPDS), if for every vertex v ∈ V − S there exists an adjacent vertex u ∈ S such that |d(u) − d(v)| ≤ 1, where d(u) and d(v) represents the degree of u and degree of v, respectively. The minimum cardinality of an EPDS of G is called the equitable power domination number (EPDN) of G, denoted by γepd (G). The vertices and edges of G are called elements. Two elements of G are neighbors if they are either incident or adjacent in G. The total graph T(G) has vertex set V(G) ∪ E(G) and two vertices of T(G) are adjacent whenever they are neighbors in G. In this paper, we obtain the EPDN of the total graph of certain graphs.

Recovery of Structural Controllability into Critical Infrastructures under Malicious Attacks

The problem of controllability of networks can be seen in critical infrastructure systems which are increasingly susceptible to random failures and/or malicious attacks. The ability to recover controllability quickly following an attack can be considered a major problem in control systems. If this is not ensured, it can enable the attacker to create more disrup-tions as well as, like the electric power networks case, violate real-time restrictions and result in the control of the network degrading and its observability reducing significantly. Thus, the present paper examines structural controllability problem that has been in focus through the equivalent problem of the Power Dominating Set (PDS) introduced in the context of electrical power network control. However, the controllability optimisation problem can be seen as computationally infeasible regarding large complex networks because such problems are considered NP-hard and as having low approximability. Hence, the ability of structural controllability recoverability will be explored as per the PDS formulation, especially following perturbations in which an attacker with sufficient knowledge of the network topology is only able to completely violate the current driver control nodes of the original control network leading to a degradation of controllability of dependent nodes. The results highlight that the use of directed Laplacian matrix can be a useful approach for analysing structural controllability of a network. The simulation results show also that an increase of a connectivity probability of the distribution of links in Directed ER networks can minimise the number of driver control nodes which is highly desirable while monitoring the entire network.

Domination and Power Domination in Certain Families of Nanostars Dendrimers

Dendrimers are hyper-branched macromolecules having various applications in diverse fields like supra-molecular chemistry, drug delivery and nanotechnology etc. The certain graph invariants such as dominating number and power dominating number can be used to characterize large number of physical properties like physio-chemical properties, thermodynamic properties, chemical and biological activities, etc. A subset of a simple undirected graph is called dominating set if every vertex of the given graph is either in that set or is adjacent to some vertex in that set. The minimum number of elements in that kind of set is called domination number. A subset of the vertex set of a graph G is said to be power dominating set (PDS) of G, if every vertex and every edge in G is observed by P. The minimum cardinality of P of a graph G is called power domination number. In this paper, the domination number and power domination number of some nanostars dendrimers have been determined.

Equitable Power Domination Number of Mycielskian of Certain Graphs

Let be a simple graph with vertex set and edge set . A set is called a power dominating set (PDS), if every vertex is observed by some vertices in by using the following rules: (i) if a vertex in is in PDS, then it dominates itself and all the adjacent vertices of and (ii) if an observed vertex in has adjacent vertices and if of these vertices are already observed, then the remaining one non-observed vertex is also observed by in . A power dominating set in is said to be an equitable power dominating set (EPDS), if for every there exists an adjacent vertex such that the difference between the degree of and degree of is less than or equal to 1, i.e., . The minimum cardinality of an equitable power dominating set of is called the equitable power domination number of and denoted by . The Mycielskian of a graph is the graph with vertex set where , and edge set In this paper we investigate the equitable power domination number of Mycielskian of certain graphs.

Resolving-power dominating sets

For a graph G(V,E) that models a facility or a multi-processor network, detection devices can be placed at vertices so as to identify the location of an intruder such as a thief or fire or saboteur or a faulty processor. Resolving-power dominating sets are of interest in electric networks when the latter helps in the detection of an intruder/fault at a vertex. We define a set S⊆V to be a resolving-power dominating set of G if it is resolving as well as a power-dominating set. The minimum cardinality of S is called resolving-power domination number. In this paper, we show that the problem is NP-complete for arbitrary graphs and that it remains NP-complete even when restricted to bipartite graphs. We provide lower bounds for the resolving-power domination number for trees and identify classes of trees that attain the lower bound. We also solve the problem for complete binary trees.

Power domination in generalized undirected de Bruijn graphs and Kautz graphs

To monitor an electric power system by placing as few phase measurement units (PMUs) as possible is closely related to the famous vertex cover problem and domination problem in graph theory. A set P is a power dominating set (PDS) of a graph G = (V, E), if every vertex and every edge in the system is observed following the observation rules of power system monitoring. The minimum cardinality of a PDS of a graph G is the power domination number γp(G). In this paper, we determine the upper bounds of power domination number of generalized undirected de Bruijn graphs and generalized undirected Kautz graphs.

Power domination in certain chemical structures

Let G(V,E) be a simple connected graph. A set S⊆V is a power dominating set (PDS) of G, if every vertex and every edge in the system is observed following the observation rules of power system monitoring. The minimum cardinality of a PDS of a graph G is the power domination number γp(G). In this paper, we establish a fundamental result that would provide a lower bound for the power domination number of a graph. Further, we solve the power domination problem in polyphenylene dendrimers, Rhenium Trioxide (ReO3) lattices and silicate networks.

Power domination with bounded time constraints

Based on the power observation rules, the problem of monitoring a power utility network can be transformed into the graph-theoretic power domination problem, which is an extension of the well-known domination problem. A set $$S$$S is a power dominating set (PDS) of a graph $$G=(V,E)$$G=(V,E) if every vertex $$v$$v in $$V$$V can be observed under the following two observation rules: (1) $$v$$v is dominated by $$S$$S, i.e., $$v \in S$$v?S or $$v$$v has a neighbor in $$S$$S; and (2) one of $$v$$v's neighbors, say $$u$$u, and all of $$u$$u's neighbors, except $$v$$v, can be observed. The power domination problem involves finding a PDS with the minimum cardinality in a graph. Similar to message passing protocols, a PDS can be considered as a dominating set with propagation that applies the second rule iteratively. This study investigates a generalized power domination problem, which limits the number of propagation iterations to a given positive integer; that is, the second rule is applied synchronously with a bounded time constraint. To solve the problem in block graphs, we propose a linear time algorithm that uses a labeling approach. In addition, based on the concept of time constraints, we provide the first nontrivial lower bound for the power domination problem.

Nodal Dynamics, Not Degree Distributions, Determine the Structural Controllability of Complex Networks

Structural controllability has been proposed as an analytical framework for making predictions regarding the control of complex networks across myriad disciplines in the physical and life sciences (Liu et al., Nature:473(7346):167–173, 2011). Although the integration of control theory and network analysis is important, we argue that the application of the structural controllability framework to most if not all real-world networks leads to the conclusion that a single control input, applied to the power dominating set, is all that is needed for structural controllability. This result is consistent with the well-known fact that controllability and its dual observability are generic properties of systems. We argue that more important than issues of structural controllability are the questions of whether a system is almost uncontrollable, whether it is almost unobservable, and whether it possesses almost pole-zero cancellations.

Power Domination in Circular-Arc Graphs

A set S⊆V is a power dominating set (PDS) of a graph G=(V,E) if every vertex and every edge in G can be observed based on the observation rules of power system monitoring. The power domination problem involves minimizing the cardinality of a PDS of a graph. We consider this combinatorial optimization problem and present a linear time algorithm for finding the minimum PDS of an interval graph if the interval ordering of the graph is provided. In addition, we show that the algorithm, which runs in ź(nlogn) time, where n is the number of intervals, is asymptotically optimal if the interval ordering is not given. We also show that the results hold for the class of circular-arc graphs.

Power domination in cylinders, tori, and generalized Petersen graphs

AbstractA set S of vertices is defined to be a power dominating set (PDS) of a graph G if every vertex and every edge in G can be monitored by the set S according to a set of rules for power system monitoring. The minimum cardinality of a PDS of G is its power domination number. In this article, we find upper bounds for the power domination number of some families of Cartesian products of graphs: the cylinders Pn□Cm for integers n ≥ 2, m ≥ 3, and the tori Cn□Cm for integers n,m ≥ 3. We apply similar techniques to present upper bounds for the power domination number of generalized Petersen graphs P(m,k). We prove those upper bounds provide the exact values of the power domination numbers if the integers m,n, and k satisfy some given relations. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 58(1), 43–49 2011

Approximation Algorithms and Hardness for Domination with Propagation

The Power Dominating Set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-size set of nodes S that power dominates all the nodes, where a node v is power dominated if (1) v is in S or v has a neighbor in S, or (2) v has a neighbor w such that w and all of its neighbors except v are power dominated. We show a hardness of approximation threshold of $2^{\log^{1-\epsilon}n}$ in contrast to the logarithmic hardness for the dominating set problem. We give an $O(\sqrt{n})$-approximation algorithm for planar graphs and show that our methods cannot improve on this approximation guarantee. Finally, we initiate the study of PDS on directed graphs and show the same hardness threshold of $2^{\log^{1-\epsilon}n}$ for directed acyclic graphs. Also we show that the directed PDS problem can be solved optimally in linear time if the underlying undirected graph has bounded tree-width.

Domination in graphs with bounded propagation: algorithms, formulations and hardness results

We introduce a hierarchy of problems between the Dominating Set problem and the Power Dominating Set (PDS) problem called the l-round power dominating set (l-round PDS, for short) problem. For l=1, this is the Dominating Set problem, and for l≥n−1, this is the PDS problem; here n denotes the number of nodes in the input graph. In PDS the goal is to find a minimum size set of nodes S that power dominates all the nodes, where a node v is power dominated if (1) v is in S or it has a neighbor in S, or (2) v has a neighbor u such that u and all of its neighbors except v are power dominated. Note that rule (1) is the same as for the Dominating Set problem, and that rule (2) is a type of propagation rule that applies iteratively. The l-round PDS problem has the same set of rules as PDS, except we apply rule (2) in “parallel” in at most l−1 rounds. We prove that l-round PDS cannot be approximated better than $2^{\log^{1-\epsilon}{n}}$ even for l=4 in general graphs. We provide a dynamic programming algorithm to solve l-round PDS optimally in polynomial time on graphs of bounded tree-width. We present a PTAS (polynomial time approximation scheme) for l-round PDS on planar graphs for $\ell=O(\frac{\log{n}}{\log{\log{n}}})$ . Finally, we give integer programming formulations for l-round PDS.

Domination in Graphs Applied to Electric Power Networks

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graphs. We consider the graph theoretical representation of this problem as a variation of the dominating set problem and define a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph G is the power domination number $\gamma_P(G)$. We show that the power dominating set (PDS) problem is NP-complete even when restricted to bipartite graphs or chordal graphs. On the other hand, we give a linear algorithm to solve the PDS for trees. In addition, we investigate theoretical properties of $\gamma_P(T)$ in trees T.