Eulerian Graphs Research Articles

Tangled Cord of FTTM4

Fuzzy Topological Topographic Mapping (FTTM) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, denoted as FTTMn, is an extension of FTTM that is arranged in a symmetrical form. The special characteristic of FTTM, namely the homeomorphisms between its components, allows the generation of new FTTM. Later, the FTTMn can also be viewed as a graph. Previously, a group of researchers defined an assembly graph and utilized it to model a DNA recombination process. Some researchers then used this to introduce the concept of tangled cords for assembly graphs. In this paper, the tangled cord for FTTM4 is used to calculate the Eulerian paths. Furthermore, it is utilized to determine the least upper bound of the Hamiltonian paths of its assembly graph. Hence, this study verifies the conjecture made by Burns et al.

A Feasibility Study on Applying the Concept of Removing Odd Nodes to Design of Water Supply Pipe Network

Objectives : In this study, the concept of the Eulerian path, which has already been proposed as a solution to the problem of head loss and residual chlorine reduction caused by complexity of nodes and pipelines, and the non-Euerian path proposed in this study were compared and their feasibility were examined.Methods : The concepts of ‘removal of odd nodes’, ‘removal of 3-intersection nodes’, and ‘removal of zero demand nodes’ were applied to the virtual water supply networks, and the number of cases were analyzed and evaluated from the viewpoint of configuration complexity and hydraulic efficiency. For Euler graph and non-Euler graph water supply networks, each total pipe length, number of removal pipes, pressure head and flow rate were simulated with EPANET2 program.Results and Discussion : As a result of simplifying and designing new output networks for virtual Euler and non-Euler graph, it was revealed that the concept of ‘removing a 3-intersection nodes’ could be the most efficient by removing more pipes. However, in terms of configuring the output network as an Euler graph, the concept of ‘removing odd nodes’ can be considered more reasonable.Conclusion : It is difficult to configuring watersupply network as a perfect Euler graph. Therefore, the application of the concept of removing odd nodes suggested in this study can promote efficient simplification of the network. In addition, it can be thought that the application of this concept can expect efficiency in terms of hydraulics and water quality.

Graph Theory in DNA Sequencing: Unveiling Genetic Patterns

Graph theory, a branch of mathematics that studies the properties and relationships of graphs, has emerged as a powerful tool in addressing the complexities of DNA sequencing. This paper highlights the application of graph theory in DNA sequencing and its implications in various aspects of genomics research. One fundamental concept in graph theory applied to DNA sequencing is the construction of the de Bruijn graph. This graph represents overlapping k-mers, subsequences of length k, as nodes, with edges connecting adjacent k-mers. By constructing and analyzing the de Bruijn graph, researchers can infer the underlying DNA sequence, detect errors, resolve repetitive regions, and identify structural variations in the genome. Graph algorithms such as Overlap path, Eulerian path and Hamiltonian path have been adapted to reconstruct complete DNA sequences from fragmented reads obtained through sequencing. These algorithms leverage the connectivity information present in the de Bruijn graph to traverse and assemble the reads, enabling the reconstruction of long DNA sequences accurately. the application of graph theory in DNA sequencing has revolutionized the field of genomics by providing powerful computational tools for DNA assembly, sequence analysis, and functional annotation. By leveraging graph theory concepts and algorithms, researchers can unravel the intricate information embedded within DNA sequences, leading to deeper insights into the genetic basis of life and its applications in various fields, including medicine, agriculture, and evolutionary biology. Future developments in graph-based algorithms and computational techniques hold promise for further enhancing our ability to unlock the secrets encoded within the vast realm of DNA sequences.

Design and Modeling Framework for DexTeR: Dexterous Continuum Tensegrity Manipulator

AbstractThe field of tensegrity faces challenges in design to facilitate efficient fabrication, and modeling due to the antagonistic nature of tension and compression elements. The research presents design methodology, and modeling framework for a human-spine inspired Dexterous continuum Tensegrity manipulatoR (DexTeR). DexTeR is a continuum manipulator that comprises of an assembly of “vertebra” modules fabricated using two curved links and 12 strings, and actuated using motor-tendon actuators. The fabrication methodology involves the construction of the equivalent graph of the module and finding the Euler path that traverses every edge of the graph exactly once. The vertices and edges of the graph correspond to the holes and strings or links of the mechanism. Unlike traditional rigid manipulators, the design results in centralization of the majority of the weight of the actuators at the base with negligible effect on the manipulator dynamics. For the first time in literature, we fabricate a tensegrity manipulator that is assembled using ten modules to conceptually validate the time and cost efficiency of the approach. A dynamic model of a vertebra module is presented using the Euler–Newton approach with screw theory representation. Each rigid link is represented using a screw, a six-dimensional vector with components of angular rotation, and linear translation. The nonlinearity in the system arises from the discontinuous behavior of the strings and the “closed-chain” nature of the mechanism. The behavior of the strings is piece-wise continuous to model their slack, compliant, or tension states.

Linear amortized time enumeration algorithms for compatible Euler trails in edge-colored graphs

Linear amortized time enumeration algorithms for compatible Euler trails in edge-colored graphs

Special Graphs of Euler’s Family* and Tracing Algorithm- (A New Approach)

There are in graph theory, some known graphs which date back from centuries. [Euler graph, Hamiltonian graph etc.] These graphs are basic roots for development of graph theory. In this paper we have discussed the novel concept of tracing Euler tour. It depends on the concept of Link vertex - a join vertex of finite number of cycles as components of Euler graph. In addition to this, a new notion of isomorphic transformation of given graph on to a given line segment known as ‘Linear Graph’ also plays an important role for tracing the Euler graph.

Implementation of Eulerian Paths based on the model of the Chinese postman: Application to the problems of garbage collection in the streets of the City of Kananga

Graph theory today plays a very important role in modeling and solving many problems. The complexity of this subject is considerable, especially since we must speak at the same time of the Chinese postman and the Eulerian journey. If it remains true that the concern is the search for the path which minimizes the cost, we understand that it will be imperative for us to glean from Fleury's algorithm, without failing to lead to the possible conditions for the realization of Euler's dreams. On the city of Kananga, whose road network is more than complex in the tangle of its arteries? In this article we seek to see how we can solve the problem of garbage collection by going through each intersection once and only once, thus minimizing cost and time.

Cycle-based formulations in Distance Geometry

The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in the given Euclidean space. Solution algorithms are generally constructed using local or global nonlinear optimization techniques. We present a new modelling technique for this problem where, instead of deciding vertex positions, the formulations decide the length of the segments representing the edges in each cycle in the graph, projected in every dimension. We propose an exact formulation and a relaxation based on a Eulerian cycle. We then compare computational results from protein conformation instances obtained with stochastic global optimization techniques on the new cycle-based formulation and on the existing edge-based formulation. While edge-based formulations take less time to reach termination, cycle-based formulations are generally better on solution quality measures.

Some properties of bases intersection graph-I

In the recent paper Iram Tahleel Jaleel Ahmed, Suryakant M Jogdand , introduced the graphical structure of vector space over a finite field  called as “Bases Intersection Graph” and discussed some basic properties of it. In the present paper, we continue the study of bases intersection graph ()   over a finite field  regarding some important properties. It is shown that the graph is regular graph, Eulerian graph etc. Further it is shown that graph is never bipartite. In the last, some properties of 2 dimensional vector space over a field of 3 elements were discussed.

Genome Sequencing Using Graph Theory Approach

Genome sequencing is the process of determining in which order the nitrogenous bases also known as nucleotides within a DNA molecule are arranged. Every organism’s genome consists of a unique sequence of nucleotides. These nucleotides bases provide the phenotypes and genotypes of a cell. In mathematics, Graph theory is the study of mathematical objects known as graphs which are made of vertices (or nodes) connected by either directed edges or indirect edges. Determining the sequence in which these nucleotides are bonded can help scientists and researchers to compare DNA between organisms, which can help show how the organisms are related. In this research, we study how graph theory plays a vital part in genome sequencing and different types of graphs used during DNA sequencing. We are going to propose several ways graph theory is used to sequence the genome. We are as well, going to explore how the graphs like Hamiltonian graph, Euler graph, and de Bruijn graphs are used to sequence the genome and advantages and disadvantages associated with each graph.

Retracted: Asymptotically Effective Method to Explore Euler Path in a Graph

Retracted: Asymptotically Effective Method to Explore Euler Path in a Graph

매듭사영의 BLL 그래프의 정의와 그 응용에 대한 연구

This paper is the research on the Knot theory in Topology. A knot is a simple closed curve in ℝ and its projection onto a plane in ℝ is called a knot projection. As the results of this paper we define a BLL(Bidirectional Linear Link) graph for a knot projection which is a bidirectional linear link representing the relations between arcs of a knot projection and obtain some properties of the BLL graphs. We also define an Eulerian cycle of the BLL graph and an Eulerian cycle of a knot projection. As the main results of this paper, we obtain the equivalent conditions of being an alternation knot projection as follows: (1) an out-degree of every vertex of the corresponding BLL graph is 2; (2) the corresponding BLL graph has an Eulerian cycle; (3) the knot projection has an Eulerian cycle. As the subsequent study, using these results of the BLL graphs, we propose the analysis on the BLL graphs for deformation operation obtaining a new alternating knot projection, decision on the tricolorability of a knot projection, and a polynomial of a knot projection.

Self-Assembly, Self-Folding, and Origami: Comparative Design Principles

Self-assembly is usually considered a parallel process while self-folding and origami are usually considered to be serial processes. We believe that these distinctions do not hold in actual experiments. Based upon our experience with 4D printing, we have developed three additional hybrid classes: (1) templated-assisted (tethered) self-assembly: e.g., when RNA is bound to viral capsomeres, the subunits are constricted in their interactions to have aspects of self-folding as well; (2) self-folding can depend upon interactions with the environment; for example, a protein synthesized on a ribosome will fold as soon as peptides enter the intracellular environment in a serial process whereas if denatured complete proteins are put into solution, parallel folding can occur simultaneously; and, (3) in turbulent environments, chaotic conditions continuously alternate processes. We have examined the 43,380 Dürer nets of dodecahedra and 43,380 Dürer nets of icosahedra and their corresponding duals: Schlegel diagrams. In order to better understand models of self-assembly of viral capsids, we have used both geometric (radius of gyration, convex hulls, angles) and topological (vertex connections, leaves, spanning trees, cutting trees, and degree distributions) perspectives to develop design principles for 4D printing experiments. Which configurations fold most rapidly? Which configurations lead to complete polyhedra most of the time? By using Hamiltonian circuits of the vertices of Dürer nets and Eulerian paths of cutting trees of polyhedra unto Schlegel diagrams, we have been able to develop a systematic sampling procedure to explore the 86,760 configurations, models of a T1 viral capsid with 60 subunits and to test alternatives with 4D printing experiments, use of MagformsTM, and origami models to demonstrate via movies the five processes described above.

A One Pass Streaming Algorithm for Finding Euler Tours

Given an undirected graph G on n nodes and m edges in the form of a data stream we study the problem of finding an Euler tour in G. Our main result is the first one-pass streaming algorithm computing an Euler tour of G in the form of an edge successor function with only mathcal O(nlog (n)) RAM, which is optimal for this setting (e.g. Sun and Woodruff (2015)). Since the output size can be much larger, we use a write-only tape to gradually output the solution. The previously best-known result for finding Euler tours in data streams is implicitly given by the W-stream algorithm of Demetrescu et al. (2010) using mathcal O(m/n) passes under the same RAM limitation. Our approach is to partition the edges into edge-disjoint cycles and to merge the cycles until a single Euler tour is achieved. In the streaming environment such a merging is far from being obvious as the limited RAM allows the processing of only a constant number of cycles at once. This enforces merging of cycles that partially are no longer present in RAM. We solve this problem with a new edge swapping technique, for which storing two certain edges per node is sufficient to merge tours without having all tour edges in RAM. The mathematical key is to model tours and their merging in an algebraic way, where certain equivalence classes represent subtours. This quite general approach might be of interest also in other routing problems.

The Lovász-Cherkassky theorem in countable graphs

Lovász and Cherkassky discovered in the 1970s independently that if G is a finite graph with a given set T of terminal vertices such that G is inner Eulerian with respect to T, then the maximal number of edge-disjoint paths connecting distinct vertices in T is ∑t∈Tλ(t,T−t) where λ is the local edge-connectivity function. The optimality of a system of edge-disjoint T-paths in the Lovász-Cherkassky theorem is witnessed by the existence of certain cuts by Menger's theorem. The infinite generalisation of Menger's theorem by Aharoni and Berger (earlier known as the Erdős-Menger Conjecture) together with the characterization of infinite Eulerian graphs due to Nash-Williams makes it possible to generalise the theorem for infinite graphs in a structural way. The aim of this paper is to formulate this generalisation and prove it for countable graphs.

Optimization of Snowplow Routes for Real-World Conditions

During the winter season, snowplowing has a significant effect on road users as it is critical to winter road maintenance and operations. The main goal of this study is to generate optimal routes for snowplowing trucks for efficient road maintenance. In addition to the conventional problem of reducing travel time and distance, this study also incorporates actual operational constraints, such as minimum maintenance standards and driver safety, to improve the overall efficiency of operations. To achieve the objectives, we first implemented the Chinese Postman Problem (CPP) to create Euler circuits from the initial routes and then identified the shortest paths by applying Dijkstra’s algorithm. Then, the Tabu search algorithm was chosen as a metaheuristic algorithm for the optimization process that finds near-optimal solutions by considering operational constraints for snowplow routes. Unsafe turning conditions and minimum maintenance standards were taken into account in the objective function defined for the optimization process. In simulations, the route obtained by our approach was compared to one with the application of CPP only in terms of travel distance, time, turning conditions, and road maintenance priority.

Alpha Labeling of Amalgamated Cycles

A graceful labeling of a bipartite graph is an \a-labeling if it has the property that the labels assigned to the vertices of one stable set of the graph are smaller than the labels assigned to the vertices of the other stable set. A concatenation of cycles is a connected graph formed by a collection of cycles, where each cycle shares at most either two vertices or two edges with other cycles in the collection. In this work we investigate the existence of \a-labelings for this kind of graphs, exploring the concepts of vertex amalgamation to produce a family of Eulerian graphs, and edge amalgamation to generate a family of outerplanar graphs. In addition, we determine the number of graphs obtained with $k$ copies of the cycle $C_n$, for both types of amalgamations.

A Feasibility Study on Application of Eulerian Path Concept to Design of Water Supply Pipe Network

Objectives : This study attempted to investigated the advantages that can be obtained by applying the concept of ‘Eulerian path’ called ‘one-touch drawing’ to the block type water supply network which actually has been operating in Korea.Methods : This study applied the ‘Eulerian path’ concept to an actual water supply network, and suggested simple C language program for searching adjacency matrix. Five Eulerian path based design scenarios were derived using the C language program. The pressure and residual chlorine distribution pattern of five design scenarios and the existing small block network were simulated using EPANET and compared each other. Also, by comparing all scenarios construction costs, the economic feasibility related to initial investment cost was reviewed.Results and Discussion : Five Eulerian path scenarios with two or more closed pipe loops were suggested using the C language program checking out adjacency matrix. One of them(scenario 5) consists of 7 closed pipes that can supply water to all nodes even if the number of pipes is reduced by about 10, and an increase in residual chlorine concentration of about 0.01 mg/L can be predicted at each node. In addition, it is expected that the initial investment cost of Scenario 5 can be reduced by about 23%, compared to the initial investment cost of the existing small block.Conclusion : Therefore, by applying the concept of an Eulerian path to the design of a water supply pipe network, it is possible to economically eliminate meaningless pipes while maintaining the redundancy of water supply.

Deriving graphs with a retracting-free bidirectional double tracing

A retracting-free bidirectional double tracing in a graph G is a closed walk which traverses every edge exactly once in each direction and such that no edge is succeeded by the same edge in the opposite direction. Studying the class Ω of all graphs admitting a retracting-free bidirectional double tracing was proposed by Ore (1951) and is, by now, of practical use to (bio)nanotechnology. In particular, this field needs various molecular polyhedra that are constructed from a single chain molecule in a retracting-free bidirectional double-tracing way. A cubic graph Q ∈ Ω has 3 h edges, where h is an odd number ≥3 . The graph of the triangular prism is the minimum cubic graph Q ∈ Ω , having 6 vertices and 9 edges. The graph of the square pyramid is the minimum polyhedral graph G in Ω , having 5 vertices and 8 edges. We analyze some possibilities for deriving new Ω -graphs from a given graph G ∈ Ω or G ∉ Ω using graph-theoretical operations. In particular, there was found that every noncycle Eulerian graph H admits a retracting-free bidirectional double tracing ( H ∈ Ω ) , which is a partial solution to the problem posed by Ore.

A graph-based path planning method for additive manufacturing of continuous fiber-reinforced planar thin-walled cellular structures

PurposeComposite cellular structures have wide application in advanced engineering fields due to their high specific stiffness and strength. As an emerging technology, continuous fiber-reinforced polymer additive manufacturing provides a cost-effective solution for fabricating composite cellular structures with complex designs. However, the corresponding path planning methods are case-specific and have not considered any manufacturing constraints. This study aims to develop a generally applicable path planning method to fill the above research gap.Design/methodology/approachThis study proposes a path planning method based on the graph theory, yielding an infill toolpath with a minimum fiber cutting frequency, printing time and total turning angle. More specifically, the cellular structure design is converted to a graph first. Then, the graph is modified to search an Eulerian path by adding an optimal set of extra edges determined through the integer linear programming method. Finally, the toolpath with minimum total turning angle is obtained with a constrained Euler path search algorithm.FindingsThe effectiveness of the proposed method is validated through the fabrication of both periodic and nonperiodic composite cellular structures, i.e. triangular unit cell-based, Voronoi diagram-based and topology optimized structures. The proposed method provides the basis for manufacturing planar thin-walled cellular structures of continuous fiber-reinforced polymer (CFRP). Moreover, the proposed method shows a notable improvement compared with the existing method. The fiber cutting frequency, printing time and total turning angle have been reduced up to 88.7%, 52.6% and 65.5%, respectively.Originality/valueA generally applicable path planning method is developed to generate continuous toolpaths for fabricating cellular structures in CFRP-additive manufacturing, which is an emerging technology. More importantly, manufacturing constraints such as fiber cutting frequency, printing time and total turning angle of fibers are considered within the process planning for the first time.