Eulerian Graphs Research Articles

Nowhere-Zero Flows on Signed Eulerian Graphs

This paper is devoted to a detailed study of nowhere-zero flows on signed Eulerian graphs. We generalize the well-known fact about the existence of nowhere-zero 2-flows in Eulerian graphs by proving that every signed Eulerian graph that admits an integer nowhere-zero flow has a nowhere-zero 4-flow. We also characterize signed Eulerian graphs with flow number 2, 3, and 4, as well as those that do not have an integer nowhere-zero flow. Finally, we discuss the existence of nowhere-zero $A$-flows on signed Eulerian graphs for an arbitrary Abelian group $A$.

Online region computations for Euler diagrams with relaxed drawing conventions

Euler diagrams are an accessible and effective visualisation of data involving simple set-theoretic relationships. Efficient algorithms to quickly compute the abstract regions of an Euler diagram upon curve addition and removal have previously been developed (the single marked point approach, SMPA), but a strict set of drawing conventions (called well-formedness conditions) were enforced, meaning that some abstract diagrams are not representable as concrete diagrams. We present a new methodology (the multiple marked point approach, MMPA) enabling online region computation for Euler diagrams under the relaxation of the drawing convention that zones must be connected regions. Furthermore, we indicate how to extend the methods to deal with the relaxation of any of the drawing conventions, with the use of concurrent line segments case being of particular importance. We provide complexity analysis and compare the MMPA with the SMPA. We show that these methods are theoretically no worse than other comparators, whilst our methods apply to any case, and are likely to be faster in practise due to their online nature. The machinery developed for the concurrency case could be of use in Euler diagram drawing techniques (in the context of the Euler Graph), and in computer graphics (e.g. the development of an advanced variation of a winged edge data structure that deals with concurrency). The algorithms are presented for generic curves; specialisations such as utilising fixed geometric shapes for curves may occur in applications which can enhance capabilities for fast computations of the algorithms' input structures. We provide an implementation of these algorithms, utilising ellipses, and provide time-based experimental data for benchmarking purposes.

When Patrolmen Become Corrupted: Monitoring a Graph Using Faulty Mobile Robots

A team of k mobile robots is deployed on a weighted graph whose edge weights represent distances. The robots move perpetually along the domain, represented by all points belonging to the graph edges, without exceeding their maximum speed. The robots need to patrol the graph by regularly visiting all points of the domain. In this paper, we consider a team of robots (patrolmen), at most f of which may be unreliable, i.e., they fail to comply with their patrolling duties. What algorithm should be followed so as to minimize the maximum time between successive visits of every edge point by a reliable patrolman? The corresponding measure of efficiency of patrolling called idleness has been widely accepted in the robotics literature. We extend it to the case of untrusted patrolmen; we denote by $${\mathfrak {I}}_k^f (G)$$ the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing $${\mathfrak {I}}_k^f (G)$$ . We investigate this problem for various classes of graphs. We design optimal algorithms for line segments, which turn out to be surprisingly different from strategies for related patrolling problems proposed in the literature. We then use these results to study general graphs. For Eulerian graphs G, we give an optimal patrolling strategy with idleness $${\mathfrak {I}}^f_k(G) = (f+1) |E|{/}k$$ , where |E| is the sum of the lengths of the edges of G. Further, we show the hardness of the problem of computing the idle time for three robots, at most one of which is faulty, by reduction from 3-edge-coloring of cubic graphs—a known NP-hard problem. A byproduct of our proof is the investigation of classes of graphs minimizing idle time (with respect to the total length of edges); an example of such a class is known in the literature under the name of Kotzig graphs.

Bounds on the price of anarchy for a more general class of directed graphs in opinion formation games

In opinion formation games with directed graphs, a bounded price of anarchy is only known for weighted Eulerian graphs. Thus, we bound the price of anarchy for a more general class of directed graphs with conditions intuitively meaning that each node does not influence the others more than she is influenced, where the bounds depend on such difference (in a ratio). We also show that there exists an example just slightly violating the conditions with an unbounded price of anarchy.

Triple Systems are Eulerian

AbstractAn Euler tour of a hypergraph (also called a rank‐2 universal cycle or 1‐overlap cycle in the context of designs) is a closed walk that traverses every edge exactly once. In this paper, using a graph‐theoretic approach, we prove that every triple system with at least two triples is eulerian, that is, it admits an Euler tour. Horan and Hurlbert have previously shown that for every admissible order >3, there exists a Steiner triple system with an Euler tour, while Dewar and Stevens have proved that every cyclic Steiner triple system of order >3 and every cyclic twofold triple system admits an Euler tour.

Experimental realization of robust dynamical decoupling with bounded controls in a solid-state spin system

We experimentally demonstrate a robust dynamical decoupling protocol with bounded controls using long soft pulses, eliminating a challenging requirement of strong control pulses in conventional implementations. This protocol is accomplished by designing the decoupling propagators to go through a Eulerian cycle of the coupler group [Phys. Rev. Lett. 90, 037901(2003)]. We demonstrate that this Eulerian decoupling scheme increases the coherence time by two orders of magnitude in our experiment under either dephasing or a universal noise environment.

Information-optimal genome assembly via sparse read-overlap graphs.

In the context of third-generation long-read sequencing technologies, read-overlap-based approaches are expected to play a central role in the assembly step. A fundamental challenge in assembling from a read-overlap graph is that the true sequence corresponds to a Hamiltonian path on the graph, and, under most formulations, the assembly problem becomes NP-hard, restricting practical approaches to heuristics. In this work, we avoid this seemingly fundamental barrier by first setting the computational complexity issue aside, and seeking an algorithm that targets information limits In particular, we consider a basic feasibility question: when does the set of reads contain enough information to allow unambiguous reconstruction of the true sequence? Based on insights from this information feasibility question, we present an algorithm-the Not-So-Greedy algorithm-to construct a sparse read-overlap graph. Unlike most other assembly algorithms, Not-So-Greedy comes with a performance guarantee: whenever information feasibility conditions are satisfied, the algorithm reduces the assembly problem to an Eulerian path problem on the resulting graph, and can thus be solved in linear time. In practice, this theoretical guarantee translates into assemblies of higher quality. Evaluations on both simulated reads from real genomes and a PacBio Escherichia coli K12 dataset demonstrate that Not-So-Greedy compares favorably with standard string graph approaches in terms of accuracy of the resulting read-overlap graph and contig N50. Available at github.com/samhykim/nsg courtade@eecs.berkeley.edu or dntse@stanford.edu Supplementary data are available at Bioinformatics online.

The Range of a Rotor Walk

In rotor walk on a graph, the exits from each vertex follow a prescribed periodic sequence. We show that any rotor walk on the d-dimensional lattice ℤd visits at least on the order of td/(d+1) distinct sites in t steps. This result extends to Eulerian graphs with a volume growth condition. In a uniform rotor walk, the first exit from each vertex is to a neighbor chosen uniformly at random. We prove a shape theorem for the uniform rotor walk on the comb graph, showing that the size of the range is of order t2/3 and the asymptotic shape of the range is a diamond. Using a connection to the mirror model, we show that the uniform rotor walk is recurrent on two different directed graphs obtained by orienting the edges of the square grid: the Manhattan lattice and the F-lattice. We end with a short discussion of the time it takes for rotor walk to cover a finite Eulerian graph.

An Eulerian Cycle Algorithm for Chinese Postman Problem

An Eulerian Cycle Algorithm for Chinese Postman Problem

Robustness of the Rotor–Router Mechanism

The rotor---router model, also called the Propp machine, was first considered as a deterministic alternative to the random walk. The edges adjacent to each node v (or equivalently, the exit ports at v) are arranged in a fixed cyclic order, which does not change during the exploration. Each node v maintains a port pointer $$\pi _v$$źv which indicates the exit port to be adopted by an agent on the conclusion of the visit to this node (the next exit port). The rotor---router mechanism guarantees that after each consecutive visit at the same node, the pointer at this node is moved to the port in the cyclic order. It is known that, in an undirected graph G with m edges, the route adopted by an agent controlled by the rotor---router mechanism eventually forms an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In Yanovski et al. (Algorithmica 37(3):165---186, 2003), it was proved that, independently of the initial configuration of the rotor---router mechanism in G, the agent locks-in in time bounded by $$2mD$$2mD, where $$D$$D is the diameter of G. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor---router mechanism. Our analysis is performed in the form of a game between a player $${\mathcal {P}}$$P intending to lock-in the agent in an Euler tour as quickly as possible and its adversary $${\mathcal {A}}$$A with the counter objective. We consider all cases of who decides the initial cyclic orders and the initial values $$\pi _v$$źv. We show, for example, that if $${\mathcal {A}}$$A provides its own port numbering after the initial setup of pointers by $${\mathcal {P}}$$P, the worst-case complexity of the lock-in problem is $${\varTheta }(m\cdot \min \{\log m,D\})$$ź(m·min{logm,D}). We also investigate the robustness of the rotor---router graph exploration in presence of faults in the pointers $$\pi _v$$źv or dynamic changes in the graph. We show, for example, that after the exploration establishes an Eulerian cycle, if k edges are added to the graph, then a new Eulerian cycle is established within $$\mathcal {O}(km)$$O(km) steps.

Closed models, strongly connected components and Euler graphs

In this paper, we continue our study of closed models defined in categories of graphs. We construct a closed model defined in the category of directed graphs which characterizes the strongly connected components. This last notion has many applications, and it plays an important role in the web search algorithm of Brin and Page, the foun-dation of the search engine Google. We also show that for this closed model, Euler graphs are particular examples of cofibrant objects. This enables us to interpret in this setting the classical result of Euler which states that a directed graph is Euleurian if and only if the in degree and the out degree of every of its nodes are equal. We also provide a cohomological proof of this last result.

Multi-Eulerian Tours of Directed Graphs

Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least once, and uses edges e and f the same number of times whenever tail(e)=tail(f). This definition leads to a simple generalization of the BEST Theorem. We then show that the minimal length of a multi-Eulerian tour is bounded in terms of the Pham index, a measure of 'Eulerianness'.

Rainbow eulerian multidigraphs and the product of cycles

An arc colored eulerian multidigraph with $l$ colors is rainbow eulerian if there is an eulerian circuit in which a sequence of $l$ colors repeats. The digraph product that refers the title was introduced by Figueroa-Centeno et al. as follows: let $D$ be a digraph and let $\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\in \Gamma$. Consider any function $h:E(D) \longrightarrow \Gamma$. Then the product $D \otimes_h \Gamma$ is the digraph with vertex set $V(D) \times V$ and $((a,x),(b,y)) \in E(D \otimes_h \Gamma)$ if and only if $(a,b) \in E(D)$ and $(x,y) \in E(h (a,b))$. In this paper we use rainbow eulerian multidigraphs and permutations as a way to characterize the $\otimes_h$-product of oriented cycles. We study the behavior of the $\otimes_h$-product when applied to digraphs with unicyclic components. The results obtained allow us to get edge-magic labelings of graphs formed by the union of unicyclic components and with different magic sums.

Algebraic Theoretic Properties of the Non-associative Class of (132)-Avoiding Patterns of AUNU Permutations: Applications in the Generation and Analysis of a General Cyclic Code

The author had in [1] and based on the report as in [2], established interplay between the adjacency matrices due to Eulerian graphs constructed by the application of AUNU numbers and the generation and analysis of a general linear code. That was achieved by constructing a [5 3 2] -linear code C of size M=8. This paper reviews such a construction of a linear code as in [1] extending the approach to a larger (linear cyclic) code which is a supper code say C1 of the [5 3 2]-linear code C of size M=8, ie C⊆ C1. To achieve this, the generator matrix G as in [1] that generated C is further developed to give a matrix say G1 which now spans a larger linear code C1 of length n=5, dimension K=4 and size M=32. This is attainable by exhausting the cyclic shifts in the rows of the matrix G to give G1. It is then shown through some existing remarks and proven theorems that the linear code generated by G1 is cyclic and has generator polynomial g(x)=1+x.

A graph theoretical equivalence for Amitsur–Levitzki theorem

Let n be a positive integer. A remarkable theorem of Amitsur–Levitzki in PI algebras states that if are any matrices with entries in any commutative ring, then , where is the standard polynomial over 2n variables . Now, let n and m be positive integers and let G be a multigraph with n vertices and m edges. Also consider a fixed labelling of edges, then any Eulerian trail in G (if exists), as a successive label of edges, corresponds to an element of with a specified parity sign. So, we may call an Eulerian trail as an odd or even Eulerian trail. R.G. Swan proved that when , then the number of even and odd Eulerian trails between any two vertices G (may be identical) are equal. We show that this latter theorem of Swan is a graph theoretical equivalence for Amitsur–Levitzki Theorem. Meanwhile, we present the Euler and Hamilton matrix of a graph with the entries denoting, respectively the number of possible Eulerian trails and Hamiltonian cycles of the graph.

A novel characterization of cubic Hamiltonian graphs via the associated quartic graphs

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1 -factor contraction. This correspondence is most useful in the case when it induces a blue and red 2 -factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian I -graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian I -graphs follows from the fact that one can choose a 1 -factor in any I -graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4 , that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected I -graphs as a subfamily.

SOLUSI PERMASALAHAN KOMPUTASI SKALA BESAR PADA SOFTWARE ADOBE FLASH DALAM PERANCANGAN MEDIA PEMBELAJARAN PENCARIAN EULERIAN CIRCUIT

Adobe Flash is a software that is used to build an interactive contents that can be attached in other things, such as presentations, games and e-learning. The Eulerian Circuit Search Instructional Media is one of examples in using Adobe Flash in building an interactive instructional media to solve graph theory problems. Visually, Adobe Flash is better than other programming software in general and it is the most suitable one for graph theory, as it uses graphics and visuals to solve problems. The weakness of Adobe Flash is when doing large scale calculations; such as when testing algorithms to find Eulerian circuits. Then an idea came about to complete the large scale calculations in Adobe Flash, which is commonly known as script time limit, with using other methodologies by changing script time limit on the settings, algorithms, partition the processes, and utulize the timers within the objects in the actionscript that handles the search algorithms of searching Eulerian circuit.

A linear time algorithm for Shortest Cyclic Cover of Strings

Merging words according to their overlap yields a superstring. This basic operation allows to infer long strings from a collection of short pieces, as in genome assembly. To capture a maximum of overlaps, the goal is to infer the shortest superstring of a set of input words. The Shortest Cyclic Cover of Strings (SCCS) problem asks, instead of a single linear superstring, for a set of cyclic strings that contain the words as substrings and whose sum of lengths is minimal. SCCS is used as a crucial step in polynomial time approximation algorithms for the notably hard Shortest Superstring problem, but it is solved in cubic time. The cyclic strings are then cut and merged to build a linear superstring. SCCS can also be solved by a greedy algorithm. Here, we propose a linear time algorithm for solving SCCS based on a Eulerian graph that captures all greedy solutions in linear space. Because the graph is Eulerian, this algorithm can also find a greedy solution of SCCS with the least number of cyclic strings. This has implications for solving certain instances of the Shortest linear or cyclic Superstring problems.

Removing and Adding Edges for the Traveling Salesman Problem

We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), we show that the approach gives a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted either to half-integral solutions to the Held-Karp relaxation or to a class of graphs that contains subcubic and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework also allows for generalizations in a natural way and leads to analogous results for the s , t -path traveling salesman problem on graphic metrics where the start and end vertices are prespecified.

Mathematical modelling and study of the encoding readout scheme for position sensitive detectors

Encoding readout methods based on different schemes have been successfully developed and tested with different types of position-sensitive detectors with strip-readout structures. However, how to construct an encoding scheme in a more general and systematic way is still under study. In this paper, we present a graph model for the encoding scheme. With this model, encoding schemes can be studied in a more systematic way. It is shown that by using an encoding readout method, a maximum of n(n−1)2+1 strips can be processed with n channels if n is odd, while a maximum of n(n−2)2+2 strips can be processed with n channels if n is even. Furthermore, based on the model, the encoding scheme construction problem can be translated into a problem in graph theory, the aim of which is to construct an Eulerian trail such that the length of the shortest subcycle is as long as possible. A more general approach to constructing the encoding scheme is found by solving the associated mathematical problem. In addition, an encoding scheme prototype has been constructed, and verified with MRPC detectors.