Hamiltonian Cycle Research Articles

Learned Upper Bounds for the Time-Dependent Travelling Salesman Problem

Given a graph whose arc traversal times vary over time, the Time-Dependent Travelling Salesman Problem consists in finding a Hamiltonian tour of least total duration covering the vertices of the graph. The main goal of this work is to define tight upper bounds for this problem by reusing the information gained when solving instances with similar features. This is customary in distribution management, where vehicle routes have to be generated over and over again with similar input data. To this aim, we devise an upper bounding technique based on the solution of a classical (and simpler) time-independent Asymmetric Travelling Salesman Problem, where the constant arc costs are suitably defined by the combined use of a Linear Program and a mix of unsupervised and supervised Machine Learning techniques. The effectiveness of this approach has been assessed through a computational campaign on the real travel time functions of two European cities: Paris and London. The overall average gap between our heuristic and the best-known solutions is about 0.001\%. For 31 instances, new best solutions have been obtained.

Resilience for loose Hamilton cycles

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum d-degree threshold for loose Hamiltonicity relative to the random k-uniform hypergraph Hk(n, p) coincides with its dense analogue whenever p≥ n−(k-1)/2+o(1). The value of p is approximately tight for d>(k+ 1)/2. This is particularly interesting because the dense threshold itself is not known beyond the cases when d≥k-2.

Joint Energy Predication and Gathering Data in Wireless Rechargeable Sensor Network

Wireless Sensor Network (WSNs) is an infrastructure-less wireless network deployed in an increasing number of wireless sensors in an ad-hoc manner. As the sensor nodes could be powered using batteries, the development of WSN energy constraints is considered to be a key issue. In wireless sensor networks (WSNs), wireless mobile chargers (MCs) conquer such issues mainly, energy shortages. The proposed work is to produce an energy-efficient recharge method for Wireless Rechargeable Sensor Network (WRSN), which results in a longer lifespan of the network by reducing charging delay and maintaining the residual energy of the sensor. In this algorithm, each node gets sorted using the K-means technique, in which the data gets distributed into various clusters. The mobile charges execute a Short Hamiltonian cycle opposite direction to reach each cluster’s anchor point. The position of the anchor points is calculated based on the energy distribution using the base station. In this case, the network will act as a spare MC, so that one of the two MCs will run out of energy before reaching the BS. After the current tours of the two MCs terminate, regression analysis for energy prediction initiates, enabling the updating of anchor points in the upcoming round. Based on the findings of the regression-based energy prediction model, the recommended algorithm could effectively refill network energy.

Comparison among Classical, Probabilistic and Quantum Algorithms for Hamiltonian Cycle Problem

Comparison among Classical, Probabilistic and Quantum Algorithms for Hamiltonian Cycle Problem

SFC Design and VNF Placement Based on Traffic Volume Scaling and VNF Dependency in 5G Networks

The development of Fifth-Generation (5G) mobile communication technology has remarkably promoted the spread of the Internet of Things (IoT) applications. As a promising paradigm for IoT, edge computing can process the amount of data generated by mobile intelligent devices in less time response. Network Function Virtualization (NFV) that decouples network functions from dedicated hardware is an important architecture to implement edge computing, deploying heterogeneous Virtual Network Functions (VNF) (such as computer vision, natural language processing, intelligent control, etc.) on the edge service nodes. With the NFV MANO (Management and Orchestration) framework, a Service Function Chain (SFC) that contains a set of ordered VNFs can be constructed and placed in the network to offer a customized network service. However, the procedure of NFV orchestration faces a technical challenge in minimizing the network cost of VNF placement due to the complexity of the changing effect of traffic volume and the dependency on the VNF relationship. To this end, we jointly optimize SFC design and VNF placement to minimize resource cost while taking account of VNF dependency and traffic volume scaling. First, the problem is formulated as an Integer Linear Programming (ILP) model and proved NP-hard by reduction from Hamiltonian Cycle problem. Then we proposed an efficient heuristic algorithm called Traffic Aware and Interdependent VNF Placement (TAIVP) to solve the problem. Compared with the benchmark algorithms, emulation results show that our algorithm can reduce network cost by 10.2% and increase service request acceptance rate by 7.6% on average.

Minimal induced subgraphs of the class of 2-connected non-Hamiltonian wheel-free graphs

Given a graph G and a graph property P we say that G is minimal with respect to P if no proper induced subgraph of G has the property P. An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph H, a graph G is H-free if G has no induced subgraph isomorphic to H. The main motivation for this paper originates from a theorem of Duffus, Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the claw-free HC-obstructions. On a similar note, Chiba and Furuya (2021), characterized all (not only the minimal) 2-connected non-Hamiltonian {K1,3,N3,1,1}-free graphs. Recently, Cheriyan, Hajebi, and two of us (2022), characterized all triangle-free HC-obstructions and all the HC-obstructions which are split graphs. A wheel is a graph obtained from a cycle by adding a new vertex with at least three neighbors in the cycle. In this paper we characterize all the HC-obstructions which are wheel-free graphs.

Hamilton cycles in bidirected complete graphs

Zaslavsky observed that the topics of directed cycles in directed graphs and alternating cycles in edge 2-colored graphs have a common generalization in the study of coherent cycles in bidirected graphs. There are classical theorems by Camion, Harary and Moser, Häggkvist and Manoussakis, and Saad which relate strong connectivity and Hamiltonicity in directed "complete" graphs and edge 2-colored "complete" graphs. We prove two analogues to these theorems for bidirected "complete" signed graphs.

On Finding Hamiltonian Cycles in Barnette Graphs

In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2–factor 𝒬 via (quasi) spanning trees of faces in G/𝒬 and study the algorithmic complexity of finding such (quasi) spanning trees of faces. Moreover, we show that if Barnette’s Conjecture is false, then hamiltonicity in 3–connected planar cubic bipartite graphs is an NP-complete problem.

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 Method for Information Security Analysis Using Information Graphs

The paper defines the concept of an information graph and proposes its use to describe information links and analyze the information flow in any organization regarding information security. Research on information graphs in this aspect should be developed, especially in identifying “dangerous graph structures” such as the Hamiltonian graphs or cliques. Information graphs containing such a structure as a Hamilton cycle can be analyzed using results from the theory of finding and existence of Hamilton cycles, an essential branch of graph theory. The paper also describes a graph of potential security conflicts, a thematic graph, i.e. a specific information graph. Two theorems concerning the properties of information graphs are also proved.

Replacement of signalized traffic network design with Hamiltonian roads: delay? Nevermind

Signal optimisation is essential in traffic engineering. The traffic light control timings should be set as optimum. However, it is challenging because the traffic network is a non-polynomial problem. In this study, the problem is evaluated from a unique perspective. The primary idea is to remove the crossings of the intersections. A Hamiltonian cycle algorithm has been used to design the network. So, vehicles are only able to join or split. Thus, no control mechanism is needed that delays or interrupts the flow. The suggested algorithm and intersection design were tested on Allsop and Charlesworth’s widely used sample network. Findings were compared with the literature in the form of delay calculated using the Highway Capacity Manual 2010 formula. The suggested network’s delay is calculated to be 98.17% and 95.45% less than the original network and recently published study-based delay, respectively. As a result, Hamiltonian roads seem sustainable in time and fuel consumption and could be used for future designs.

Development and operation of the Hemispherical Blackbody (HSBB) for the calibration of infrared radiometers with a hemispherical acceptance angle.

Longwave downward radiation measurements are fundamental for investigating the surface and atmospheric energy budget of the Earth. Such measurements are performed using broadband infrared radiometers that have a hemispherical acceptance angle. A new hemispherical reference blackbody has been developed to provide a new traceability path of longwave downward radiation measurements to the SI. The Hemispherical Blackbody (HSBB) was specifically designed to meet the large opening angle of radiometers used for measuring longwave downward radiation. The first step was to find a suitable design candidate for the HSBB by simulating the effective emissivity using the Monte Carlo ray-tracing software STEEP. The most promising candidate was then realized in two versions with similar coatings and both versions were taken into operation.

Analysis of the whole genome traceability and transmission path simulation experiment of the local cluster COVID-19 epidemic

Objective: To trace and characterize the whole genome of SARS-CoV-2 of confirmed cases in the outbreak of COVID-19 on July 31, 2021 in Henan Province. Method: Genome-wide sequencing and comparative analysis were performed on positive nucleic acid samples of SARS-CoV-2 from 167 local cases related to the epidemic on July 31, 2021, to analyze the consistency and evolution of the whole genome sequence of virus. Results: Through high-throughput sequencing, a total of 106 cases of SARS-CoV-2 whole genome sequences were obtained. The results of genome analysis showed that the whole genome sequences of 106 cases belonged to the VOC/Delta variant strain (B.1.617.2 clade), and the whole genome sequences of 106 cases were shared with the genomes of 3 imported cases from Myanmar admitted to a hospital in Zhengzhou. On the basis of 45 nucleotide sites, 1-5 nucleotide variation sites were added, and the genome sequence was highly homologous. Conclusion: Combined with the comprehensive analysis of viral genomics, transmission path simulation experiments and epidemiology, it is determined that the local new epidemic in Henan Province is caused by imported cases in the nosocomial area, and the spillover has caused localized infection in the community. At the same time, it spills over to some provincial cities and results in localized clustered epidemics.

The Complexity of Routing Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs of consecutive edges on the walk are permitted. Forbidden-transition graphs and related models have found applications in a variety of fields, such as routing in optical telecommunication networks, road networks, and bio-informatics. A widely-studied special case are edge-colored graphs, where a compatible walk is forbidden to take two edges of the same color in a row. We initiate the study of fundamental problems on finding paths, cycles and walks in forbidden-transition graphs from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters. Among several results, we prove that finding a simple compatible path between given endpoints in a forbidden-transition graph is W[1]-hard when parameterized by the vertex-deletion distance to a linear forest (so it is also hard when parameterized by pathwidth or treewidth). On the other hand, we show an algebraic trick that yields tractability when parameterized by treewidth for finding a compatible Hamiltonian cycle in the edge-colored graph setting.

Identification Conditions for the Solvability of NP-Complete Problems for the Class of Prefractal Graphs

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, and cryptocurrency networks) are distinguished by their multielement nature and the dynamics of connections between their elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems lead to an additional complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamic graphs that are used to model the structures of network system, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes prefractal graphs. This article investigates NP-complete problems on prefractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, and an independent set. The conditions under which it is possible to obtain an answer about the existence of and construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions for some problems are identified.

On asymptotic packing of geometric graphs

A set of geometric graphs is geometric-packable if it can be asymptotically packed into every sequence of drawings of the complete graph K n . For example, the set of geometric triangles is geometric-packable due to the existence of Steiner Triple Systems. When G is the 4-cycle (or 4-cycle with a chord), we show that the set of plane drawings of G is geometric-packable. In contrast, the analogous statement is false when G is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions). A convex geometric graph is convex-packable if it can be asymptotically packed into the convex drawings of the complete graphs. For each planar Hamiltonian graph G , we determine whether or not a plane G is convex-packable. Many of our proofs explicitly construct these packings; in these cases, the packings exhibit a symmetry that mirrors the vertex transitivity of K n .

Making Graphs Solvable in Peg Solitaire

Abstract: Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. Since then peg solitaire has been considered on quite a few classes of graphs. Beeler and Gray introduced the natural idea of adding edges to make an unsolvable graph solvable. Recently, the graph invariant ms(G), which is the minimal number of additional edges needed to make G solvable, has been introduced and investigated on banana trees by the authors. In this article, we determine ms(G) for several families of unsolvable graphs. Furthermore, we provide some general results for this number of Hamiltonian graphs and graphs obtained via binary graph operations.

Distributed transformations of Hamiltonian shapes based on line moves

We consider a discrete system of n simple indistinguishable devices, called agents, forming a connected shape SI on a two-dimensional square grid. Agents are equipped with a linear-strength mechanism, called a line move, by which an agent can push a whole line of consecutive agents in one of the four cardinal directions in a single time-step. We study the problem of transforming an initial shape SI into a given target shape SF via a finite sequence of line moves in a distributed model, where each agent can observe the states of nearby agents in a Moore neighbourhood. We develop the first distributed connectivity-preserving transformation that exploits line moves. The transformation solves the line formation problem. That is, starting from any shape SI whose associated graph contains a Hamiltonian path known to them, the agents can form a final straight line SL. The complexity of the transformation is O(nlog2⁡n) moves, which is asymptotically equivalent to that of the best-known centralised transformations.

Spectrum of Johnson graphs

In this paper we prove new bounds on the second eigenvalue of Johnson graphs. We then apply these bounds to obtain new results on the modularity of Johnson graphs and their random subgraphs, hamiltonicity of Johnson graphs and thresholds of the appearance of the Hamilton cycles and giant components. Furthermore, we provide general bounds on vertex connectivity and the stability of the modularity for (n,d,λ)-graphs.

A Hamiltonian Path-Based Enciphering Technique with the use of a Self-Invertible Key Matrix

Objective: The symmetric encryption technique is one of the most important fields for securing communications between people. In order to produce complex ciphertext, we are introducing the new enciphering technique with the help of the Hamiltonian path, a self-invertible key matrix for encryption and decryption. Methods: There are many kinds of symmetric enciphering methods, like the Caesar Cipher, Atbash Cipher, Hill Cipher, etc. All these methods use a common key for encryption, and while decrypting the inverse of that matrix should be found, it is also too hard to share the common key. To reduce this terminology, we proposed the novel enciphering method with the help of a self-invertible key matrix. Findings: We are using the generated selfinvertible key matrix as a key matrix; the degree of the self-invertible matrix is even. If our graph does not form an even-degree adjacency matrix, then we have to make it into an even-degree adjacency matrix by adding dummy edges. Novelty: As we are using a self-invertible matrix as a key matrix, we do not need to compute the inverse of the key matrix for the process of decryption. This helps us to reduce the complexity of finding the inverse while decrypting the original message. Keywords: Graph Theory Encryption; Hill Cipher; Hamiltonian Path; Adjacency Matrix; Self Invertible Matrix