scholarly journals Computation of the Double Metric Dimension in Convex Polytopes

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Liying Pan ◽  
Muhammad Ahmad ◽  
Zohaib Zahid ◽  
Sohail Zafar

A source detection problem in complex networks has been studied widely. Source localization has much importance in order to model many real-world phenomena, for instance, spreading of a virus in a computer network, epidemics in human beings, and rumor spreading on the internet. A source localization problem is to identify a node in the network that gives the best description of the observed diffusion. For this purpose, we select a subset of nodes with least size such that the source can be uniquely located. This is equivalent to find the minimal doubly resolving set of a network. In this article, we have computed the double metric dimension of convex polytopes R n and Q n by describing their minimal doubly resolving sets.

Author(s):  
Antonio Padilla-Meléndez ◽  
Ana Rosa Del Aguila-Obra

Today, everyone recognizes that we live in the so-called knowledge society. In this society, new possibilities based on and around IT and the Internet arise for human beings. IT technology has also made the organizations where they work change rapidly as well as the wider general business environment. The development of the Internet in the early 1990s was both the catalyst and an example of this phenomenon. This computer network allowed the development of social networks, or virtual communities of people who use these networks to communicate and to collaborate. We shall concentrate on the specific changes that have taken place in the workplace because of the introduction and increased usage of IT.


Electronics ◽  
2019 ◽  
Vol 8 (5) ◽  
pp. 587
Author(s):  
Taewon Min ◽  
Changhee Joo

We investigate the problem of source detection in information spreading throughout a densely-connected network. Previous works have been developed mostly for tree networks or applied the tree-network results to non-tree networks assuming that the infection occurs in the breadth first manner. However, these approaches result in low detection performance in densely-connected networks, since there is a substantial number of nodes that are infected through the non-shortest path. In this work, we take a two-step approach to the source detection problem in densely-connected networks. By introducing the concept of detour nodes, we first sample trees that the infection process likely follows and effectively compare the probability of the sampled trees. Our solution has low complexity of O ( n 2 log n ) , where n denotes the number of infected nodes, and thus can be applied to large-scale networks. Through extensive simulations including practical networks of the Internet autonomous system and power grid, we evaluate our solution in comparison with two well-known previous schemes and show that it achieves the best performance in densely-connected networks.


2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhi-Bo Zheng ◽  
Ashfaq Ahmad ◽  
Zaffar Hussain ◽  
Mobeen Munir ◽  
Muhammad Imran Qureshi ◽  
...  

For a graph G , an ordered set S ⊆ V G is called the resolving set of G , if the vector of distances to the vertices in S is distinct for every v ∈ V G . The minimum cardinality of S is termed as the metric dimension of G . S is called a fault-tolerant resolving set (FTRS) for G , if S \ v is still the resolving set ∀ v ∈ V G . The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of G . Due to enormous application in science such as mathematics and computer, the notion of the resolving set is being widely studied. In the present article, we focus on determining the FTMD of a generalized wheel graph. Moreover, a formula is developed for FTMD of a wheel and generalized wheels. Recently, some bounds of the FTMD of some of the convex polytopes have been computed, but here we come up with the exact values of the FTMD of two families of convex polytopes denoted as D k for k ≥ 4 and Q k for k ≥ 6 . We prove that these families of convex polytopes have constant FTMD. This brings us to pose a natural open problem about the existence of a polytope having nonconstant FTMD.


Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Hua Wang ◽  
Muhammad Azeem ◽  
Muhammad Faisal Nadeem ◽  
Ata Ur-Rehman ◽  
Adnan Aslam

In computer networks, vertices represent hosts or servers, and edges represent as the connecting medium between them. In localization, some special vertices (resolving sets) are selected to locate the position of all vertices in a computer network. If an arbitrary vertex stopped working and selected vertices still remain the resolving set, then the chosen set is called as the fault-tolerant resolving set. The least number of vertices in such resolving sets is called the fault-tolerant metric dimension of the network. Because of the variety of applications of the metric dimension in different areas of sciences, many generalizations were proposed, and fault tolerant is one of them. In this paper, we computed the fault-tolerant metric dimension of triangular snake, ladder, Mobius ladder, and hexagonal ladder networks. It is important to observe that, in all these classes of networks, the fault-tolerant metric dimension and metric dimension differ by 1.


2013 ◽  
Vol 4 (2) ◽  
Author(s):  
Rudolf Maresch

Durch den digitalen Medienwandel ist der Begriff der Öffentlichkeit problematisch geworden. Die Debatte fokussiert sich zumeist auf die Frage, ob die sogenannte bürgerliche Öffentlichkeit durch das Internet im Niedergang begriffen ist oder eine Intensivierung und Pluralisierung erfährt. Rudolf Maresch zeichnet die berühmte Untersuchung der Kategorie durch Jürgen Habermas nach und zieht den von ihm konstatierten Strukturwandel der Öffentlichkeit in Zweifel. Dagegen verweist er auf die gouvernementalen und medialen Prozesse, die jede Form von Kommunikation immer schon gesteuert haben. Öffentlichkeit sei daher ein Epiphänomen nicht allein des Zeitungswesens, sondern der bereits vorgängig ergangenen postalischen Herstellung einer allgemeinen Adressierbarkeit von Subjekten. Heute sei Öffentlichkeit innerhalb der auf Novitäts- und Erregungskriterien abstellenden Massenmedien ein mit anderen Angeboten konkurrierendes Konzept. Mercedes Bunz konstatiert ebenfalls eine Ausweitung und Pluralisierung von Öffentlichkeit durch den digitalen Medienwandel, sieht aber die entscheidenden Fragen in der Konzeption und Verteilung von Evaluationswissen und Evaluationsmacht. Nicht mehr die sogenannten Menschen, sondern Algorithmen entscheiden über die Verbreitung und Bewertung von Nachrichten. Diese sind in der Öffentlichkeit – die sie allererst erzeugen – weitgehend verborgen. Einig sind sich die Autoren darin, dass es zu einer Pluralisierung von Öffentlichkeiten gekommen ist, während der Öffentlichkeitsbegriff von Habermas auf eine singuläre Öffentlichkeit abstellt. </br></br>Due to the transformation of digital media, the notion of “publicity” has become problematic. In most cases, the debate is focused on the question whether the internet causes a decline of so-called civic publicity or rather intensifies and pluralizes it. Rudolf Maresch outlines Jürgen Habermas's famous study of this category and challenges his claim concerning its “structural transformation,” referring to the governmental and medial processes which have always already controlled every form of communication. Publicity, he claims, is an epiphenomenon not only of print media, but of a general addressability of subjects, that has been produced previously by postal services. Today, he concludes, publicity is a concept that competes with other offers of mass media, which are all based on criteria of novelty and excitement. Mercedes Bunz also notes the expansion and pluralization of the public sphere due to the change of digital media, but sees the crucial issues in the design and distribution of knowledge and power by evaluation. So-called human beings no longer decide on the dissemination and evaluation of information, but algorithms, which are for the most part concealed from the public sphere that they produce in the first place. Both authors agree that a pluralization of public sphere(s) has taken place, while Habermas's notion of publicity refers to a single public sphere.


Author(s):  
Jia-Bao Liu ◽  
Muhammad Faisal Nadeem ◽  
Mohammad Azeem

Aims and Objective: The idea of partition and resolving sets plays an important role in various areas of engineering, chemistry and computer science such as robot navigation, facility location, pharmaceutical chemistry, combinatorial optimization, networking, and mastermind game. Method: In a graph to obtain the exact location of a required vertex which is unique from all the vertices, several vertices are selected this is called resolving set and its generalization is called resolving partition, where selected vertices are in the form of subsets. Minimum number of partitions of the vertices into sets is called partition dimension. Results: It was proved that determining the partition dimension a graph is nondeterministic polynomial time (NP) problem. In this article, we find the partition dimension of convex polytopes and provide their bounds. Conclusion: The major contribution of this article is that, due to the complexity of computing the exact partition dimension we provides the bounds and show that all the graphs discussed in results have partition dimension either less or equals to 4, but it cannot been be greater than 4.


2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.


Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.


Author(s):  
Bojan Ljuijić

Beside the fact that the Internet was not primarily educational network (it didn’t emerge from the intention to be systematically used in the field of education), shortly after it emerged, possibilities of its application in education were recognised. This paper is dedicated to analysis of the most important chronological moments (technological and social in the first place) that were crucial in sense of comprehensive application of the Internet in service of education in general, but also in service of adult education. Having all mentioned in focus, in more details, we analysed emergence and development of the Internet observed as educational computer network in frame of general development of information and communication technologies. While realising mentioned analysis, our focus was on four historical periods of educational computer technologies. We also intended to emphasize the activities of international institutions that followed, encouraged and supported the development of the Internet use and the use of other information and communication technologies in the field of education. According to that, we distinguished the main moments referring activities of these organisations which describe in the best manner their contributions to growing application of the Internet in education in general, but also in adult education.


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