Structure and Substructure Connectivity of Hypercube-Like Networks

2020 ◽  
Vol 30 (03) ◽  
pp. 2040007
Author(s):  
Cheng-Kuan Lin ◽  
Eddie Cheng ◽  
László Lipták
Keyword(s):  
Text Structure ◽  
Network Architectures ◽  
Connected Subgraph ◽  
Menger's Theorem ◽  
The Family ◽  
Minimum Number ◽  
Structure Formula ◽  
Connected Subgraphs ◽  
Twisted Cubes

The connectivity of a graph [Formula: see text], [Formula: see text], is the minimum number of vertices whose removal disconnects [Formula: see text], and the value of [Formula: see text] can be determined using Menger’s theorem. It has long been one of the most important factors that characterize both graph reliability and fault tolerability. Two extensions to the classic notion of connectivity were introduced recently: structure connectivity and substructure connectivity. Let [Formula: see text] be isomorphic to any connected subgraph of [Formula: see text]. The [Formula: see text]-structure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] such that every element of [Formula: see text] is isomorphic to [Formula: see text], and the removal of [Formula: see text] disconnects [Formula: see text]. The [Formula: see text]-substructure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] whose removal disconnects [Formula: see text] and every element of [Formula: see text] is isomorphic to a connected subgraph of [Formula: see text]. The family of hypercube-like networks includes many well-defined network architectures, such as hypercubes, crossed cubes, twisted cubes, and so on. In this paper, both the structure and substructure connectivity of hypercube-like networks are studied with respect to the [Formula: see text]-star [Formula: see text] structure, [Formula: see text], and the [Formula: see text]-cycle [Formula: see text] structure. Moreover, we consider the relationships between these parameters and other concepts.

THE IMPORTANCE OF THE FAMILY FOR GENEALOGICAL STRUCTURE OF VOLYN BEEF OF CATTLE

2016 ◽  
Vol 52 ◽  
pp. 82-94 ◽  
Author(s):  
A. Ye. Pochukalin
Keyword(s):  
Beef Cattle ◽  
Live Weight ◽  
Breeding Value ◽  
Farm Families ◽  
Cattle Breeding ◽  
Reproductive Ability ◽  
The Family ◽  
Minimum Number ◽  
Related Group ◽  
Economic Use

One of the ways of increasing level of animal economically useful traits is selection work with farm families. In pedigree cattle breeding of Ukraine families are a statistical component of breed genealogy. Among the main scientific works on working with families, it should be noted minimum number of female ancestors, proposed by D. T. Vinnichuk, to determine the breeding value, different categories, classification and techniques for evaluating related groups of females. The aim of our research was to analyse importance of farm families for genealogical structure of the breed. The research was on basis of data of primary breeding records at the herd of Volyn Beef cattle of “Zorya” breeding farm, Kovel district, Volyn region. Akula 102, Galka 37 and Galka 1537 families belonging to Krasavchyk 3004 bloodline, Smorodyna 613, Korona 2382 and Visla 1016 families – Tsebryk 3888 bloodline, Kalyna 212, Verba 1536 and Garna 536 families – Yamb 3066 bloodlines, Kazka 433, Galka 421 and Bystra 1124 families – Buinyi 3042 bloodline, Rozetka 1313, Arfa 599 and Bulana 943 families – Sonnyi-Kaktus 3307-9828 bloodline, and Palma 275, Desna 870 and Veselka 444 families – Mudryi 9100 bloodline were characterized. Belonging to a bloodline was determined by the father's side of female ancestors. Structural units of families: branches, branching with identifying the best individuals on breeding traits were submitted to identify the best combinations and successful use of closely related breeding. Comparing assessment of related groups of females on the main breeding traits belonging to Krasavchyk 3004 bloodline, it was noted that the cows of Akula 102 family predominated in live weight at 5 years’ age, milk ability and economic use duration, whereas the cows of Galka 1537 family – on traits of reproductive ability. Smorodyna 613 family of Tsebryk 3888 bloodline had high duration of economic use and cows’ live weight at 5 years’ age compared with Korana 2382 and Visla 1016 families with equal values of the exterior traits (height measures) and coefficient of reproductive ability. The families of Mudryi 9100 bloodline in terms of reproduction (calving interval, coefficient of reproductive ability) had the highest figures of cows’ milk ability and live weight. The cows of Bulana 943 family had a considerable predominance over representatives of Rozetka 1313 and Arfa 599 families of Sonnyi-Kaktus 3307-9828 bloodline by main economically useful traits. High indices of reproductive ability were noted in these families. Heifers of the families of Buinyi 3042 bloodline had high live weight at 18 months’ age at average values of milk ability and cows’ live weight at 5 years’ age. More equal figures of growth rate, exterior and economic use duration were observed in the cows of Kalyna 212, Verba 1536 and Garna 536 families of Yamb 3066 bloodline. Breeding by families in beef cattle breeding is an important element of selection, because it allows to evaluate not only related group of female ancestor, but also to analyse a successful combination with lines and purposeful use of closely related breeding by the best representatives of a breed.

Nullification of Torus knots and links

2014 ◽  
Vol 23 (11) ◽  
pp. 1450058 ◽  
Author(s):  
Claus Ernst ◽  
Anthony Montemayor
Keyword(s):  
Torus Knots ◽  
The Family ◽  
Minimum Number ◽  
Knots And Links

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. In this paper we investigate the nullification numbers of a particular knot family, namely the family of torus knots and links.

scholarly journals EFEKTIVITAS PROGRAM KELUARGA HARAPAN DALAM UPAYA PENGENTASAN KEMISIKINAN DI KECAMATAN LEMBANG KABUPATEN BANDUNG BARAT

2018 ◽  
Vol 2 (2) ◽  
pp. 175-189
Author(s):  
Fitri Irvanasari
Keyword(s):  
Qualitative Approach ◽  
Literature Study ◽  
Teaching Techniques ◽  
Key Informants ◽  
Family Program ◽  
The Family ◽  
Minimum Number ◽  
Descriptive Method

The research title is "The Effectiveness of the Hope Family Program in Poverty Control Efforts in Lembang District, West Bandung Regency". The problem in this research is the ineffectiveness of the Family Hope Program (PKH) in Lembang District. In this study, using a descriptive method with a qualitative approach. Teaching techniques through literature study, interviews with informants and use of documents. The key informants in this study were the Lembang PKH Assistance Coordinator, as well as supporting informants consisting of UPPKH staff in West Bandung Regency, the Secretary of Lembang District and several PKH participants in Lembang District. Based on the results of the research that the implementation of PKH has been able to reduce the burden on poor community PKH participants, and is able to increase PKH participant participation in accessing health and education, however, there are still some obstacles that show that PKH implementation is less than optimal, this is evidenced by PKH being right on target causing problems , PKH funds that are not in accordance with the circumstances of PKH participants and PKH facilitators, the minimum number of mentors, and disbursement of PKH funds that are not on time. Efforts to overcome these obstacles include updating data, increasing PKH funds, increasing counterpart quota, and extending the time for distribution of funds.

Total k-rainbow reinforcement number in graphs

2020 ◽  
pp. 2050101
Author(s):  
L. Shahbazi ◽  
H. Abdollahzadeh Ahangar ◽  
R. Khoeilar ◽  
S. M. Sheikholeslami
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
The Family ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Complete Bipartite Graphs ◽  
Dominating Function

Let [Formula: see text] be an integer, and let [Formula: see text] be a graph. A k-rainbow dominating function (or [Formula: see text]RDF) of [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the family of all subsets of [Formula: see text] such that for very [Formula: see text] with [Formula: see text], the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a [Formula: see text]RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. A k-rainbow dominating function [Formula: see text] in a graph with no isolated vertex is called a total k-rainbow dominating function if the subgraph of [Formula: see text] induced by the set [Formula: see text] has no isolated vertices. The total k-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of the total [Formula: see text]-rainbow dominating function on [Formula: see text]. The total k-rainbow reinforcement number of [Formula: see text], denoted by [Formula: see text], is the minimum number of edges that must be added to [Formula: see text] in order to decrease the total k-rainbow domination number. In this paper, we investigate the properties of total [Formula: see text]-rainbow reinforcement number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total [Formula: see text]-rainbow reinforcement number of some classes of graphs including paths, cycles and complete bipartite graphs.

A New Algorithm to Solve the Maximal Connected Subgraph Problem Based on Parallel Molecular Computing

2016 ◽  
Vol 13 (10) ◽  
pp. 7692-7695
Author(s):  
Nan Guo ◽  
Jun Pu ◽  
Zhaocai Wang ◽  
Dongmei Huang ◽  
Lei Li ◽  
...  
Keyword(s):  
Dna Computing ◽  
Combinatorial Problems ◽  
Search Problem ◽  
Connected Subgraph ◽  
Complete Problems ◽  
Portfolio Problem ◽  
Path Search ◽  
Np Complete ◽  
Connected Subgraphs ◽  
Connected Vertex

DNA computing is widely used in complex NP-complete problems, such as the optimal portfolio problem, the optimum path search problem. DNA computing, having the characteristics of high parallelism, huge storage capacity and low energy loss, is very suitable for solving complex combinatorial problems. The maximal connected subgraph problem aims to find a connected vertex subset with maximal number of vertices in a simple undirected graph. Using biological computing technology, we give a new DNA algorithm to solve the maximal connected subgraphs problem with O(n) time complexity, which can significantly reduce the complexity of computing compared with the previous algorithms.

scholarly journals Sequential Spiking Neural P Systems with Local Scheduled Synapses without Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Alia Bibi ◽  
Fei Xu ◽  
Henry N. Adorna ◽  
Francis George C. Cabarle
Keyword(s):  
Computational Models ◽  
P Systems ◽  
Dynamic Graphs ◽  
Computational Power ◽  
Spiking Neural P Systems ◽  
Graphs And Networks ◽  
The Family ◽  
Minimum Number ◽  
Sequential Mode ◽  
Maximum Minimum

Spiking neural P systems with scheduled synapses are a class of distributed and parallel computational models motivated by the structural dynamism of biological synapses by incorporating ideas from nonstatic (i.e., dynamic) graphs and networks. In this work, we consider the family of spiking neural P systems with scheduled synapses working in the sequential mode: at each step the neuron(s) with the maximum/minimum number of spikes among the neurons that can spike will fire. The computational power of spiking neural P systems with scheduled synapses working in the sequential mode is investigated. Specifically, the universality (Turing equivalence) of such systems is obtained.

The restrained k-rainbow reinforcement number of graphs

2020 ◽  
pp. 2150026
Author(s):  
Saeed Shaebani ◽  
Saeed Kosari ◽  
Leila Asgharsharghi
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
The Family ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
General Graphs

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple graph. A restrained [Formula: see text]-rainbow dominating function (R[Formula: see text]RDF) of [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the family of all subsets of [Formula: see text], such that every vertex [Formula: see text] with [Formula: see text] satisfies both of the conditions [Formula: see text] and [Formula: see text] simultaneously, where [Formula: see text] denotes the open neighborhood of [Formula: see text]. The weight of an R[Formula: see text]RDF is the value [Formula: see text]. The restrained[Formula: see text]-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an R[Formula: see text]RDF of [Formula: see text]. The restrained[Formula: see text]-rainbow reinforcement number [Formula: see text] of [Formula: see text], is defined to be the minimum number of edges that must be added to [Formula: see text] in order to decrease the restrained [Formula: see text]-rainbow domination number. In this paper, we determine the restrained [Formula: see text]-rainbow reinforcement number of some special classes of graphs. Also, we present some bounds on the restrained [Formula: see text]-rainbow reinforcement number of general graphs.

scholarly journals A Branch-and-Price Framework for Decomposing Graphs into Relaxed Cliques

2020 ◽  
Author(s):  
Timo Gschwind ◽  
Stefan Irnich ◽  
Fabio Furini ◽  
Roberto Wolfler Calvo
Keyword(s):  
Social Network ◽  
Graph Partitioning ◽  
Disease Spread ◽  
Computational Studies ◽  
Branch And Price ◽  
Unified Framework ◽  
Covering Problems ◽  
Structure Preserving ◽  
The Family ◽  
Minimum Number

We study the family of problems of partitioning and covering a graph into/with a minimum number of relaxed cliques. Relaxed cliques are subsets of vertices of a graph for which a clique-defining property—for example, the degree of the vertices, the distance between the vertices, the density of the edges, or the connectivity between the vertices—is relaxed. These graph partitioning and covering problems have important applications in many areas such as social network analysis, biology, and disease-spread prevention. We propose a unified framework based on branch-and-price techniques to compute optimal decompositions. For this purpose, new, effective pricing algorithms are developed, and new branching schemes are invented. In extensive computational studies, we compare several algorithmic designs, such as structure-preserving versus dichotomous branching, and their interplay with different pricing algorithms. The final chosen branch-and-price setup produces results that demonstrate the effectiveness of all components of the newly developed framework and the validity of our approach when applied to social network instances.

scholarly journals On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2441
Author(s):  
Michal Staš
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Symmetric Graphs ◽  
The Family ◽  
Minimum Number ◽  
The Common ◽  
First Time ◽  
Edge Crossings

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main purpose of this paper is to determine the crossing numbers of the join products of six symmetric graphs on six vertices with paths and cycles on n vertices. The idea of configurations is generalized for the first time onto the family of subgraphs whose edges cross the edges of the considered graph at most once, and their lower bounds of necessary numbers of crossings are presented in the common symmetric table. Some proofs of the join products with cycles are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs.

scholarly journals Fault-Tolerant quantum computation with constant overhead

2014 ◽  
Vol 14 (15&16) ◽  
pp. 1339-1371
Author(s):  
Daniel Gottesman
Keyword(s):  
Fault Tolerant ◽  
Quantum Circuit ◽  
Error Correcting Codes ◽  
Parity Check ◽  
Low Density Parity Check ◽  
The Family ◽  
Minimum Number ◽  
Logical Qubits ◽  
Quantum Error ◽  
Parity Check Codes

What is the minimum number of extra qubits needed to perform a large fault-tolerant quantum circuit? Working in a common model of fault-tolerance, I show that in the asymptotic limit of large circuits, the ratio of physical qubits to logical qubits can be a constant. The construction makes use of quantum low-density parity check codes, and the asymptotic overhead of the protocol is equal to that of the family of quantum error-correcting codes underlying the fault-tolerant protocol.

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