Rotation Graph Research Articles

DBTRG: De Bruijn Trim rotation graph encoding for reliable DNA storage

DNA is a high-density, long-term stable, and scalable storage medium that can meet the increased demands on storage media resulting from the exponential growth of data. The existing DNA storage encoding schemes tend to achieve high-density storage but do not fully consider the local and global stability of DNA sequences and the read and write accuracy of the stored information. To address these problems, this article presents a graph-based De Bruijn Trim Rotation Graph (DBTRG) encoding scheme. Through XOR between the proposed dynamic binary sequence and the original binary sequence, k-mers can be divided into the De Bruijn Trim graph, and the stored information can be compressed according to the overlapping relationship. The simulated experimental results show that DBTRG ensures base balance and diversity, reduces the likelihood of undesired motifs, and improves the stability of DNA storage and data recovery. Furthermore, the maintenance of an encoding rate of 1.92 while storing 510 KB images and the introduction of novel approaches and concepts for DNA storage encoding methods are achieved.

Diameter Estimates for Graph Associahedra

Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. We give a tight bound of $\Theta(m)$ on the diameter of trivially perfect graph associahedra on $m$ edges. We consider the maximum diameter of associahedra of graphs on $n$ vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. We also prove that the maximum diameter of associahedra of graphs of pathwidth two is $\Theta (n\log n)$. Finally, we give the exact diameter of the associahedra of complete split and of unbalanced complete bipartite graphs.

Distributions of restricted rotation distances

Rotation distances measure the differences in structure between rooted ordered binary trees. The one-dimensional skeleta of associahedra are rotation graphs, where two vertices representing trees are connected by an edge if they differ by a single rotation. There are no known efficient algorithms to compute rotation distance between trees and thus distances in rotation graphs. Limiting the allowed locations of where rotations are permitted gives rise to a number of notions related to rotation distance. Allowing rotations at a minimal such set of locations gives restricted rotation distance. There are linear-time algorithms to compute restricted rotation distance, where there are only two permitted locations for rotations to occur. The associated restricted rotation graph has an efficient distance algorithm. There are linear upper and lower bounds on restricted rotation distance with respect to the sizes of the reduced tree pairs. Here, we experimentally investigate the expected restricted rotation distance between two trees selected at random of increasing size and find that it lies typically in a narrow band well within the earlier proven linear upper and lower bounds.

Simulation to Determine The Amount of Fan Rotation on a Cooling Machine Using The Fuzzy Logic Method

This Fan Rotation Simulation Application on Cooling Machines with Fuzzy Logic is an application that can simulate the fan rotation process based on the input voltage and temperature given by the user. In carrying out this simulation, this application uses a stress index with a category of small, medium, medium, high or higher. Temperature index with categories cooler, cooler, warmer, warmer and hotter. While the fan speed index uses the very slow, slow, rather fast, fast and very fast categories. This system can analyze the amount of fan rotation in RPM (Rotation Per Minute) automatically based on the input voltage and temperature and is equipped with a graphic facility that displays the simulation results of fan rotation, namely temperature graphs and fan rotation graphs. However. The designed system is equipped with fan rotation speed animation, to simulate fan rotation in the temperature range of -50 C.

Synthesis method for planetary gear trains without using rotation graphs

The displacement graphs (d-graphs) are the final form of planetary gear trains (PGTs) synthesis. The existing configuration synthesis methods for the d-graphs of PGTs need rotation graphs (r-graphs) as the intermediate step and produce a large amount of isomorphic KCs. This paper proposes a method to obtain d-graphs directly from the geared graph. Meanwhile, a uniform isomorphism identification method suitable for parent graphs, geared graphs, and d-graphs is presented. Finally, a fully automatic method based on parent graphs is tested on synthesizing 1-DOF PGTs with up to nine links. Our synthesis results and those in the literature are comparatively analyzed, and the reasons for contradictory synthesis results are analyzed.

Bounds on the Diameter of Graph Associahedra

Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph G encodes the combinatorics of search trees on G, defined recursively by a root r together with search trees on each of the connected components of G − r. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length n, the classical associahedra, is 2n - 6 for a large enough n. We give a tight bound of Θ(m) on the diameter of trivially perfect graph associahedra on m edges. We consider the maximum diameter of associahedra of graphs on n vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ(n log n).

Automatic structural synthesis of non-fractionated 2-DOF planetary gear trains

Both 1-DOF and multi-DOF planetary gear trains (PGTs) have extensive application in various kinds of mechanical equipment. However, the existing synthesis methods mainly focused on 1-DOF PGTs, while the research on the synthesis of multi-DOF PGTs is very limited. This paper presents an automatic method to synthesize non-fractionated 2-DOF PGTs, free of degenerate and isomorphic structures. First, non-fractionated parent graphs corresponding to each link assortment are synthesized. Second, the transfer vertex equation is established to synthesize 2-DOF rotation graphs. A method based on a set of independent loops is presented to precisely detect disconnected and fractionated graphs including parent graphs and rotation graphs. Finally, edge level equation is established to synthesize 2-DOF displacement graphs. The atlas of non-fractionated 2-DOF PGTs with up to nine links is automatically generated. The synthesis results of 8- and 9-link 2-DOF PGTs, to the best of our knowledge, are new results that have not been reported in literature.

Similarity recognition and isomorphism identification of planar kinematic chains

Similarity recognition and isomorphism identification (SRII) of kinematic chains (KCs) are key problems facing their innovative regenerative design and configuration synthesis. Although various detection methods have been proposed, most of them are based on topological characteristics or some codings, which are only suitable for single type KC and error identification is generated in some cases. In this study, a set of SRII methods is proposed based on the graph theory definition of similarity and isomorphism. These are appropriate for planar single and multiple joints KCs, planetary gear trains (PGTs), contracted graphs, and multicolor graphs. The computational complexity and effectiveness of the algorithm are assessed on 10-bar, 15-bar, 21-bar, 28-bar, and 42-bar single-joint KCs, 4-bar tricolor topological graphs, and 8-bar contracted graphs. All the results are in agreement with those in the cited literature, and the advantage in isomorphism identification efficiency of the multibar single-hinged KC is evident. Finally, SRII are performed for all 8-bar one-degree-of-freedom (1-DOF) and 9-bar 2-DOF planar single-hinged KCs, all 8-bar 1-DOF KCs with two multiple joints, and all 6-bar 1-DOF rotation graphs through identification program, with the aim to provide a reference for the application of the algorithm.

Enumeration of displacement graphs of epicyclic gear train from a given rotation graph using concept of building of kinematic units

Use of graph theory for enumeration of Epicyclic gear trains (EGTs) is common during conceptual stage. According to theory of structural synthesis of EGTs, two kinds of graphs are defined namely rotation and displacement graphs, for a given number of links and one degree of freedom. In case, all the non-isomorphic rotation graphs are enumerated, then the next stage of synthesis involves the generation of displacement graphs for each rotation graph. To this end, given a rotation graph, the present work proposes an algorithm for generation of all the displacement graphs. Firstly, the rotation graph is decomposed into modules called kinematic units (KUs). Then, for each KU, all the edge-labeling possibilities of turning pairs are evaluated. Finally, displacement graphs are obtained by building the edge labeled KUs in all possible ways. Acyclic graph based representation is employed and displacement graphs with and up to 7 links are enumerated. The enumeration of displacement graphs is treated as two different cases corresponding to rotation graphs enumerated using recursive and non-recursive schemes and the results are in agreement with C.H. Hsu and others.

The Complete Set of One-Degree-of-Freedom Planetary Gear Trains With Up to Nine Links

The structural synthesis of planetary gear trains (PGTs) is helpful for innovating transmission systems in machinery. A great deal of research has been devoted to the synthesis of one-degree-of-freedom (1-DOF) PGTs over the past half century. However, most synthesis methods are limited to PGTs with no more than eight links. Moreover, the synthesis results are not consistent with each other. Until now, the inconsistency of synthesis results is still unresolved and exact synthesis results remain elusive. This paper presents a systematic and fully automatic method based on parent graphs to synthesize 1-DOF PGTs. The complete database of rotation graphs (r-graphs) and displacement graphs (d-graphs) of 1-DOF PGTs with up to nine links is established for the first time. All possible reasons for the contradictory synthesis results in the literature are analyzed and the controversy in the existing synthesis results which has lasted for nearly half a century is completely resolved. The exact results of the 6-, 7-, and 8-link r-graphs are confirmed to be 27, 152, and 1070, respectively. The exact results of the 6-, 7-, and 8-link d-graphs are confirmed to be 81, 647, and 6360, respectively. Additionally, the new results of 8654 r-graphs and 71,837 d-graphs of 9-link PGTs are provided for the first time.

On the Diameter of Tree Associahedra

We consider a natural notion of search trees on graphs, which we show is ubiquitous in various areas of discrete mathematics and computer science. Search trees on graphs can be modified by local operations called rotations, which generalize rotations in binary search trees. The rotation graph of search trees on a graph $G$ is the skeleton of a polytope called the graph associahedron of $G$.We consider the case where the graph $G$ is a tree. We construct a family of trees $G$ on $n$ vertices and pairs of search trees on $G$ such that the minimum number of rotations required to transform one search tree into the other is $\Omega (n\log n)$. This implies that the worst-case diameter of tree associahedra is $\Theta (n\log n)$, which answers a question from Thibault Manneville and Vincent Pilaud. The proof relies on a notion of projection of a search tree which may be of independent interest.

New Graph Representation for Planetary Gear Trains

Planetary gear trains (PGTs) are widely used in machinery to transmit angular velocity ratios or torque ratios. The graph theory has been proved to be an effective tool to synthesize and analyze PGTs. This paper aims to propose a new graph model, which has some merits relative to the existing ones, to represent the structure of PGTs. First, the rotation graph and canonical rotation graph of PGTs are defined. Then, by considering the edge levels in the rotation graph, the displacement graph and canonical displacement graph are defined. Each displacement graph corresponds to a PGT having the specified functional characteristics. The synthesis of five-link one degree-of-freedom (1DOF) PGTs is used as an example to interpret and demonstrate the applicability of the present graph representation in the synthesis process. The present graph representation can completely avoid the generation of pseudo-isomorphic graphs and can be used in the computer-aided synthesis and analysis of PGTs.

Detection of Degenerate Structure in Single Degree-of-Freedom Planetary Gear Trains

Graph theory is a powerful tool in structural synthesis and analysis of planetary gear trains (PGTs). In this paper, a new algorithm has been developed for detecting degenerate structure in planetary gear trains. The proposed algorithm is based on the concept of fundamental circuits' rotation graphs. Detection of degeneracy is entirely based on finding one key element. The key element or link that makes planetary gear train into two groups is found in this work. The main advantage of the proposed method lies in the drastic reduction in the required combinatorial analysis compared to other methods available.

Topological Synthesis of Epicyclic Gear Trains Using Vertex Incidence Polynomial

Epicyclic gear trains (EGTs) are used in the mechanical energy transmission systems where high velocity ratios are needed in a compact space. It is necessary to eliminate duplicate structures in the initial stages of enumeration. In this paper, a novel and simple method is proposed using a parameter, Vertex Incidence Polynomial (VIP), to synthesize epicyclic gear trains up to six links eliminating all isomorphic gear trains. Each epicyclic gear train is represented as a graph by denoting gear pair with thick line and transfer pair with thin line. All the permissible graphs of epicyclic gear trains from the fundamental principles are generated by the recursive method. Isomorphic graphs are identified by calculating VIP. Another parameter “Rotation Index” (RI) is proposed to detect rotational isomorphism. It is found that there are six nonisomorphic rotation graphs for five-link one degree-of-freedom (1-DOF) and 26 graphs for six-link 1-DOF EGTs from which all the nonisomorphic displacement graphs can be derived by adding the transfer vertices for each combination. The proposed method proved to be successful in clustering all the isomorphic structures into a group, which in turn checked for rotational isomorphism. This method is very easy to understand and allows performing isomorphism test in epicyclic gear trains.

Magnetic Simulation and Analysis of Radial Flux Permanent Magnet Generator using Finite Element Method

This paper discusses magnetic simulation and analysis of radial flux permanent magnet generator (PMG) using finite element method (FEM) by utilizing open source software FEMM 4.2. The specification of generator is 25 V, 28 A, 3 phase, 300 rpm. The analyzed magnetic flux was in the air gap, stator teeth and slots to find out the distribusian pattern and its fluctuation. The simulations were conducted in no-load and nominal load (28 A) conditions. Furthermore the maximum flux density of simulation (B g(sim) ) was used to calculate phase voltage E ph to find out the magnitude of generated electromotive force (EMF). The calculation results were presented as voltage vs. rotation graph in no-load condition and voltage vs. current graph in nominal load condition. Both graphs were validated with E ph of experiment result (E ph(exp) ) and E ph that the value of B g obtained from analytical calculation (E ph(calc) ). The final results showed that in no-load condition, E ph graph with B g(sim) (E ph(sim) ) was close to E ph(exp) and E ph(calc). The error rate with respect to the experiment was 6,9%. In nominal load condition, E ph(sim) graph almost coincides with E ph(calc.) graph, with the voltage drop of both was 0,441 V. Both graphs however were far different from E ph(exp) graph, which has 9 V of voltage drop. The overall results demonstrated that magnetic distribution pattern presented by FEM was very helpful to avoid magnetic flux accumulation in a particular segment. Besides B g(sim) facilitated to predict the value of E ph .

The Edge Rotation Graph

Let P be a set of n points on the plane in general position, n ? 3. The edge rotation graph $${\mathcal{E} \mathcal{R} \mathcal{G}(P,k)}$$ of P is the graph whose vertices are the plane geometric graphs on P with exactly k edges, two of which are adjacent if one can be obtained from the other by an edge rotation. In this paper we study some structural properties of $${\mathcal{E} \mathcal{R} \mathcal{G}(P,k)}$$ , such as its connectivity and diameter. We show that if the vertices of $${\mathcal{E} \mathcal{R} \mathcal{G}(P,k)}$$ are not triangulations of P, then it is connected and has diameter O(n 2). We also show that the chromatic number of $${\mathcal{E} \mathcal{R} \mathcal{G}(P,k)}$$ is O(n), and show how to compute an implicit coloring of its vertices. We also study edge rotations in edge-labelled geometric graphs.

MINIMUM FEEDBACK ARC SETS IN ROTATOR AND INCOMPLETE ROTATOR GRAPHS

A feedback vertex/arc set (abbreviated as FVS/FAS) of a graph is a subset of the vertices/arcs which contains at least one vertex/arc for every cycle of that graph. A minimum FVS/FAS is an FVS/FAS which contains the smallest number of vertices/arcs. Hsu et al. [11] first proposed an algorithm which can find a minimum FVS in a rotator graph. In this paper, we present a formula for finding an FAS for a rotator graph and prove that the FAS is minimum. This formula can be easily implemented by an efficient algorithm which obtains a minimum FAS in a rotator graph. Finally, we also present a concise formula for finding a minimum FAS in an incomplete rotator graph in this paper.

Wavelength Rotation Graph Algorithm Based on Physical Impairment Constraints in Optical Networks

As the network is converting to transparent optical networks, the impact of physical layer impairments on transmission performance can't be ignored. The impairments will decrease the quality of signals. To find a lightpath with high quality of transmission (QoT), this paper adds impairment constraints to the routing and wavelength algorithm based on wavelength rotation graph (WRG-RWA) and proposes impairment aware WRG-RWA algorithm. The constraints consider amplified spontaneous emission noises and polarization mode dispersion which are the most important liner impairments. The new algorithm estimates the impact of the impairments after the route is selected, and selects another route when the previous one fails in comparing with the constraints. By simulating, the new algorithm has preferably improvement on blocking probability and wavelength efficiency by contrast with the WRG-RWA algorithm and the impairment aware FF algorithm. The simulation results show the new algorithm can effectively decrease the blocking probability of transmission and improve the performance of optical networks.

Hamilton Cycles in Restricted and Incomplete Rotator Graphs

The nodes of a rotator graph are the permutations of n, and an arc is directed from u to v if the rst r symbols of u can be rotated one position to the left to obtain v. Restricted rotator graphs restrict the allowable rotations to r2 R for some Rf 2; 3;:::;ng. Incomplete rotator graphs only include nodes whose nal symbol is i m for a xed maximum value m2 f1; 2;:::;ng. Restricted rotator graphs are directed Cayley graphs, whereas incomplete rotator graphs are not Cayley graphs. Hamilton cycles exist for rotator graphs (Corbett 1992), restricted rotator graphs with R =fn 1;ng (Ruskey and Williams 2010), and incomplete rotator graphs for all m (Ponnuswamy and Chaudhary 1994). These previous results are based on sequence building operations that we name ‘reusing’, ‘recycling’, and ‘rewinding’. In this article, we combine these operations to create Hamilton cycles in rotator graphs that are (1) restricted by R =f2; 3;ng, (2) restricted by R = f2; 3;n 1;ng and incomplete for any m, and (3) restricted by R =fn 2;n 1;ng and incomplete for any m. Result (1) is ‘optimal’ since restricted rotator graphs are not strongly connected for R =f3;ng when n is odd, and do not have Hamilton cycles for R =f2;ng when n is even (Rankin 1944, Swan 1999). Similarly, we prove (3) is ‘optimal’. Our Hamilton cycles can be easily implemented for potential applications, and we provide O(1)-time algorithms that generate successive rotations for (1){(3).

An efficient algorithm for minimum feedback vertex sets in rotator graphs

For a rotator graph with n! nodes, Hsu and Lin [C.C. Hsu, H.R. Lin, H.C. Chang, K.K. Lin, Feedback Vertex Sets in Rotator Graphs, in: Lecture Notes in Comput. Sci., vol. 3984, 2006, pp. 158–164] first proposed an algorithm which constructed a feedback vertex set (FVS) with time complexity O ( n n − 3 ) . In addition, they found that the size of the FVS is n ! / 3 , which was proved to be minimum. In this paper, we present an efficient algorithm which constructs an FVS for a rotator graph in O ( n ! ) time and also obtains the minimum FVS size n ! / 3 . In other words, this algorithm derives the optimal result with linear time complexity in terms of the number of nodes in the rotator graph.