Eulerian Graphs Research Articles

A circle criterion for a generalized cross graph in terms of minimal excluded minors

Geelen and Oum described classes of minimal excluded pivot-minors for a simple graph to be a circle graph and for a delta-matroid to be Eulerian. Pivot-equivalence classes of circle simple graphs and delta-matroids arise in the investigation of Eulerian cycles on cross graphs (4-valent graphs with cross structure). The results established by Geelen and Oum rely on some lemmas in their work, which are shown below to be not quite correct. We consider generalized cross graphs, which arise in the description of rotating circuits on cross graphs. For such graphs we derive a circle criterion: we reproduce and augment the arguments due to Geelen and Oum, and we improve some incorrectly formulated statements. As a result, we obtain the same list of 166 inequivalent graphs, the minimal excluded minors for a generalized cross graph to be a circle graph. Bibliography: 14 titles.

Transition Polynomial as a Weight System for Binary Delta-Matroids

To a singular knot K with n double points, one can associate a chord diagram with n chords. A chord diagram can also be understood as a 4-regular graph endowed with an oriented Euler circuit. L. Traldi introduced a polynomial invariant for such graphs, called a transition polynomial. We specialize this polynomial to a multiplicative weight system, that is, a function on chord diagrams satisfying 4-term relations and determining thus a finite type knot invariant. We prove a similar statement for the transition polynomial of general ribbon graphs and binary delta-matroids defined by R. Brijder and H. J. Hoogeboom, which defines, as a consequence, a finite type invariant of links.

Asymptotically Effective Method to Explore Euler Path in a Graph

Euler path is one of the most interesting and widely discussed topics in graph theory. An Euler path (or Euler trail) is a path that visits every edge of a graph exactly once. Similarly, an Euler circuit (or Euler cycle) is an Euler trail that starts and ends on the same node of a graph. A graph having Euler path is called Euler graph. While tracing Euler graph, one may halt at arbitrary nodes while some of its edges left unvisited. In this paper, we have proposed some precautionary steps that should be considered in exploring a deadlock-free Euler path, i.e., without being halted at any node. Simulation results show that our proposed approach improves the process of exploring the Euler path in an undirected connected graph without interruption. Furthermore, our proposed algorithm is complete for all types of undirected Eulerian graphs. The paper concludes with the proofs of the correctness of proposed algorithm and its computation complexity.

3-manifolds represented by 4-regular graphs with three Eulerian cycles

3-manifolds represented by 4-regular graphs with three Eulerian cycles

COMBINATORIAL RECONSTRUCTION OF TWO-DIMENSIONAL WORDS IN THE HYPOTHESIS SHIFT 1

The article considers the formulation of the problem of reconstruction of two-dimensional words by a given multiset of subwords, under the hypothesis that this subset is generated by the displacement of a two-dimensional window of fixed size by an unknown two-dimensional word with a shift 1. A variant of the combinatorial solution of this reconstruction problem is proposed, based on a two-fold application of the one-dimensional word reconstruction method using the search for Eulerian paths or cycles in the de Bruyne multiorgraph. The efficiency of the method is discussed under the conditions of a square two-dimensional shift window one having a large linear size.

An Ideal-Based Dot Total Graph of a Commutative Ring

In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail.

Usage of Eulerian and Hamiltonian Graph in Pandemic Situation

The existence of Euler and Hamiltonian graph make it easier to solve a real-life problem. During the time of pandemic “Covid-19”, it is very essential for each one of us to be vaccinated. Vaccination is done in the hospitals by using Eulerian and Hamiltonian graphs not only to prevent people from infecting but also to increase the speed of vaccination. In this paper, the authors have discussed the use of graph theory to effectively handle the covid – 19 pandemic situations in the allocation of patients to appropriate hospitals. The work done in this paper may be utilized for the hospital–patient management as well as for the study by upcoming researchers to utilize the graph theory in any complicated situations.

Weld line recognition and path planning with spherical tank inspection robots

AbstractPeriodic inspections are required for the safe operation of large pressure vessels such as spherical tanks. Inspection robots have been applied in large pressure vessels due to their low cost and high efficiency. This paper presents a robotic system for the inspection of spherical tanks, which can identify and track weld lines on the shortest running route. Two‐dimensional (2D) weld maps were prepared for robot path planning on the basis of the actual distribution of weld lines. In 2D weld maps, indispensable repetitive lines were added to form an Eulerian circuit that traversed all weld lines. In addition, an improved Fleury algorithm was proposed to solve Eulerian circuit and plan an optimal running route for robot inspection. To accurately identify weld lines, deep learning networks were constructed and trained with weld line data sets, which were captured by the camera mounted in the front of the robot. The laboratory experiments indicated that the inspection robot could identify weld lines within 0.2–0.25 s and track weld lines with a maximum offset of ±20 mm. The experiment results demonstrated that the robot could plan the shortest path to traverse all weld lines on the experimental platform. In the field tests, the virtual simulation of weld path planning on spherical tanks was explored in detail. The field tests of a spherical tank (3000 m3) verified that the robotic system could improve the efficiency and stability of inspection operations and replace manual inspection with automated weld line recognition and weld path planning.

Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem . Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O (log n / log log n ) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, Mądry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem.

Shannon, Euler, and Mazes

One of Claude Shannon’s best remembered “toys” was his maze-solving machine, created by partitions on a rectangular grid. A mechanical mouse was started at one point in the maze with the task of finding cheese at another point. Relays under the board guided successive moves, each of which were taken in the first open counterclockwise direction from the previous move. In belated honor of Shannon’s centenary and of amnesia in the mouse at age 70, we compare this deterministic search strategy with a random search requiring no memory. For simplicity, the rectangular grid with partitions is replaced by a finite connected graph. A maze is then a graph with some given destination node. The worst case required number of steps to find the cheese for deterministic searches and the expected number for random searches are remarkably similar, each being, for example, $|\mathcal {E}^2|$|E2| taken over all graphs of $|\mathcal {E}|$|E| edges. Finally, we demonstrate a simple improvement to the above algorithms that generates an Eulerian cycle on the directed edges of $G$G, i.e., a walk on $G$G of $2|\mathcal {E}|$2|E| steps that traverses each edge in $G$G exactly once in each direction before returning to the starting point.

Eulerian and Even-Face Graph Partial Duals

Eulerian and bipartite graph is a dual symmetric concept in Graph theory. It is well-known that a plane graph is Eulerian if and only if its geometric dual is bipartite. In this paper, we generalize the well-known result to embedded graphs and partial duals of cellularly embedded graphs, and characterize Eulerian and even-face graph partial duals of a cellularly embedded graph by means of half-edge orientations of its medial graph.

A Waveform-Encoded SAR Implementation Using a Limited Number of Cyclically Shifted Chirps

Synthetic aperture radar (SAR) provides high-resolution images of the Earth’s surface irrespective of sunlight and weather conditions. In conventional spaceborne SAR, nadir echoes caused by the pulsed operation of SAR may significantly affect the SAR image quality. Therefore, the pulse repetition frequency (PRF) is constrained within the SAR system design to avoid the appearance of nadir echoes in the SAR image. As an alternative, the waveform-encoded SAR concept using a pulse-to-pulse variation of the transmitted waveform and dual-focus postprocessing can be exploited for nadir echo removal and to alleviate the PRF constraints. In particular, cyclically shifted chirps have been proposed as a possible waveform variation scheme. However, a large number of distinct waveforms is required to enable the simple implementation of the concept. This work proposes a technique based on the Eulerian circuit for generating a waveform sequence starting from a reduced number of distinct cyclically shifted chirps that can be effectively exploited for waveform-encoded SAR. The nadir echo suppression performance of the proposed scheme is analyzed through simulations using real TerraSAR-X data and a realistic nadir echo model that shows how the number of distinct waveforms and therefore the system complexity can be reduced without significant performance loss. These developments reduce the calibration burden and make the concept viable for implementation in future SAR systems.

On the degeneracy of spin ice graphs, and its estimate via the Bethe permanent

The concept of spin ice can be extended to a general graph. We study the degeneracy of spin ice graph on arbitrary interaction structures via graph theory. We map spin ice graphs to the Ising model on a graph and clarify whether the inverse mapping is possible via a modified Krausz construction. From the gauge freedom of frustrated Ising systems, we derive exact, general results about frustration and degeneracy. We demonstrate for the first time that every spin ice graph, with the exception of the one-dimensional Ising model, is degenerate. We then study how degeneracy scales in size, using the mapping between Eulerian trails and spin ice manifolds, and a permanental identity for the number of Eulerian orientations. We show that the Bethe permanent technique provides both an estimate and a lower bound to the frustration of spin ices on arbitrary graphs of even degree. While such a technique can also be used to obtain an upper bound, we find that in all finite degree examples we studied, another upper bound based on Schrijver inequality is tighter.

Find subtrees of specified weight and cycles of specified length in linear time

AbstractWe apply the Euler tour technique to find subtrees of specified weight as follows. Let such that and , where . Let be a tree of vertices and let be vertex weights such that and for all . We prove that a subtree of of weight exists and can be found in linear time. We apply it to show, among others, the following: Every planar hamiltonian graph with minimum degree has a cycle of length for every with . Every 3‐connected planar hamiltonian graph with and even has a cycle of length or .Each of these cycles can be found in linear time if a Hamilton cycle of the graph is given. This study was partially motivated by conjectures of Bondy and Malkevitch on cycle spectra of 4‐connected planar graphs.

Anti-Eulerian digraphs

Let D be a digraph. A closed anti-directed Euler trail of D is a closed trail in which consecutive arcs have opposite directions and each arc of D occurs exactly once. A digraph is anti-Eulerian if it contains a closed anti-directed Euler trail. A tournament T is anti-Eulerian if and only if both d+(v) and d−(v) are even for any v∈V(T). The Cartesian product of two directed cycles Cn1→□Cn2→ is anti-Eulerian if and only if gcd(n1,n2)=1. If n1,n2,…,n2k can be partitioned into k relatively prime pairs, then Cn1→□Cn2→□⋯□Cn2k→ is anti-Eulerian. If each of Di (1≤i≤k) is an anti-Eulerian digraph and at least one of them is not a bipartite digraph, then D1□D2□⋯□Dk is anti-Eulerian.

Graph Theory for Primary School Students with High Skills in Mathematics

Graph theory is a powerful representation and problem-solving tool, but it is not included in present curriculum at school levels. In this study we perform a didactic proposal based in graph theory, to provide students useful and motivational tools for problem solving. The participants, who were highly skilled in mathematics, worked on map coloring, Eulerian cycles, star polygons and other related topics. The program included six sessions in a workshop format and four creative sessions where participants invented their own mathematical challenges. Throughout the experience they applied a wide range of strategies to solve problems, such as look for a pattern, counting strategies or draw the associated graph, among others. In addition, they created as challenges the same type of problems posed in workshops. We conclude that graph theory successfully increases motivation of participants towards mathematics and allows the appearance and enforcement of problem-solving strategies.

JCB Call for Papers for the Special Issue on “40 Years of Computational Biology: In Honor of Professor Michael Waterman's 80th Birthday”

<i>JCB</i> Call for Papers for the Special Issue on “40 Years of Computational Biology: In Honor of Professor Michael Waterman's 80th Birthday”

Some properties of open subset intersection graph of a topological space

In the recent paper, authors introduced a graph topological structure, called open subset intersection graph of a topological space ϒ(τ) on a finite set X. In this present paper, we continue the study of the graph ϒ(τ) on a finite set X. It is shown that, if τ is a discrete topology on X and |X| ≥ 3, then the graph ϒ(τ) is not bipartite and regular. If τ is a discrete topology on X with |X| = 3 then it is shown that the ϒ(τ) is an edge-transitive graph, Perfect graph, Eulerian and Hamiltonian graph . Moreover, we detremine exact value of the independence number, vertex connectivity and edge connectivity of the graph ϒ(τ).

A two-stage optimization approach for inspection plan formulation of comprehensive inspection train: The China case

The regular and dynamic inspections of high-speed railways (HSR) are carried out by comprehensive inspection train (CIT). Generally, the network of HSR (e.g., the HSR network in China) has some obvious characteristics such as complex network structure, high traffic density and strict maintenance restrictions of CIT, which make the formulation of CIT inspection plan extremely hard. However, formulating and optimizing the CIT inspection plan is one of the key issues to ensure the safety of HSR operations and to improve the efficiency of testing equipment. In this paper, the original model is proposed for the multiple CITs inspection plan formulation problem (MCIT-IPFP) firstly, and it is proved that the problem belongs to the NP-hard class of computational complexity by reduction. By decomposing the problem into two stages, one is used to divide the HSR network evenly for each CIT, and the other one is used to generate the path, the timetable, and maintenance base assignment for each CIT under requirements of reducing the total time cost and the difficulty of solving. In order to improve the efficiency of calculation, a Constructive Heuristic Algorithm with feedback regulation based on the greedy algorithm is designed to generate the task division plan. With the purpose of quickly formulating the inspection plans with low time cost for each CIT with Euler circuit characteristics, a simulated annealing algorithm with efficient neighborhood Euler circuit generation strategy is proposed. Finally, a case study is conducted to test the efficiency of the algorithm by taking the China’s HSR network as the background. Compared with the previous manual inspection plan, the utilization rate of CITs will increase by 30%-50% when using the models and algorithms proposed in this paper.

Proof of a conjecture on the Wiener index of Eulerian graphs

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. A connected graph is Eulerian if its vertex degrees are all even. In Gutman et al. (2014) the authors proved that the cycle is the unique graph maximising the Wiener index among all Eulerian graphs of given order. They also conjectured that for Eulerian graphs of order n≥26 the graph consisting of a cycle on n−2 vertices and a triangle that share a vertex is the unique Eulerian graph with second largest Wiener index. The conjecture is known to hold for all n≤25 with exception of six values. In this paper we prove the conjecture.