Longest Path Research Articles

Visualizing and Bibliometrically Analyzing Information and Communication Technology-Based Blended Learning

With the rapid development of information and communication technologies, blended learning has gained popularity after the COVID-19 pandemic. This study aims to visualize and bibliometrically analyze information and communication technology-based blended learning. Through bibliometric analyses and visualization via both VOSviewer and CitNetExplorer, this study identified the top ten cited authors, organizations, countries, sources, and keywords in the field of information and communication technologies used for blended learning, providing meaningful references for future research. Most studies reported that information and communication technology-based blended learning could improve learning effectiveness and environments despite numerous challenges revealed by several previous studies. In addition, this study suggested the main research methods to explore ICT-assisted blended learning given the present meta-analytical methods in the longest path where a citation network was established. Future research directions were suggested.

Bonds Intersecting Long Paths in \(k\) -Connected Graphs

In 1966, Gallai asked whether there is a vertex which passes through all longest paths of a connected graph. Although this has been verified for some special classes of graphs such as outerplanar graphs, circular arc graphs, and series-parallel graphs, the answer is negative for general graphs. In this paper, we prove among other results that if we replace the vertex by a bond, then the answer is affirmative. A bond of a graph is a minimal nonempty edge-cut. In particular, in any 2-connected graph, the set of all edges incident to a vertex is a bond, called a vertex-bond. Clearly, for a 2-connected graph, a path passes through a vertex if and only if it meets the vertex-bond with respect to . Therefore, a very natural approach to Gallai’s question is to study whether there is a bond meeting all longest paths. Let denote the length of a longest path of connected graphs. We show that there is a bond meeting all paths of length at least and for any 2- and 3-connected graph, respectively. For any -connected graph , we show that there is a bond meeting all paths of length at least , where if is even and if is odd. Our results also provide analogs of the results on bonds meeting long cycles given in [P.-L. Wu, Combin. Probab. Comput., 6 (1997), pp. 107–113; and S. McGuinness, Combinatorica, 25 (2005), pp. 439–450].

Conflict-tolerant and conflict-free multi-agent meeting

In the Multi-Agent Meeting problem (MAM), the task is to find the optimal meeting location for multiple agents, as well as a path for each agent to that location. Among all possible meeting locations, the optimal meeting location has the minimum cost according to a given cost function. Two cost functions are considered in this research: (1) the sum of all agents paths' costs to the meeting location (SOC) and (2) the cost of the longest path among them (MKSP). MAM has many real-life applications, such as choosing a gathering point for multiple traveling agents (humans, cars, or robots).In this paper, we divide MAM into two variants. In its basic version, MAM allows multiple agents to occupy the same location, i.e., it is conflict tolerant. For MAM, we introduce MM*, a Multi-Directional Heuristic Search algorithm, that finds the optimal meeting location under different cost functions. MM* generalizes the Meet in the Middle (MM) bidirectional search algorithm to the case of finding an optimal meeting location for multiple agents. Several admissible heuristics are proposed for MM*, and experiments demonstrate the benefits of MM*.As agents may be embodied in the world, a solution to MAM may contain conflicting paths, where more than one agent occupies the same location at the same time. The second variant of the MAM problem is called Conflict-Free Multi-Agent Meeting (CF-MAM), where the task is to find the optimal meeting location for multiple agents (as in MAM) as well as conflict-free paths (in the same manner as the prominent Multi-Agent Path Finding problem (MAPF)) to that location. For optimally solving CF-MAM, we introduce two novel algorithms, which combine MAM and MAPF solvers. We prove the optimality of both algorithms and compare them experimentally, showing the pros and cons of each algorithm.1

Detours in directed graphs

We study two “above guarantee” versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least distG(s,t)+k. Bezáková et al. [7] proved that on undirected graphs the problem is fixed-parameter tractable (FPT). Our first main result establishes a connection between Longest Detour on directed graphs and 3- Disjoint Paths on directed graphs. Using these new insights, we design a 2O(k)⋅nO(1) time algorithm for the problem on directed planar graphs. Furthermore, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k. We obtain dichotomy results about Longest Path above Diameter on undirected and directed graphs.

Degree conditions and relative length of longest paths and cycles in graphs

The relative length of a graph G, denoted by diff(G), is diff(G)=p(G)−c(G), where p(G) and c(G) denote the orders of a longest path and cycle, respectively. Let G be a 3-connected graph of order n. In this paper, we show that diff(G)≤1 if the maximum degree of any two nonadjacent vertices of G is more than (n+1)/3, and the lower bound is sharp. Moreover, if G is 2-connected graph satisfying the degree condition above, then diff(G)≤1 or G is in a special family of graphs.

The maximum number of cliques in graphs with prescribed order, circumference and minimum degree

Erdős determined the maximum size of a nonhamiltonian graph of order n and minimum degree at least k in 1962. Recently, Ning and Peng generalized Erdős’s work and gave the maximum number of s-cliques hs(n,c,k) of graphs with prescribed order n, circumference c and minimum degree at least k. But for some quadruples n,c,k,s the maximum number of s-cliques is not attained by a graph of minimum degree k. For example, h2(15,14,3)=77 is attained by a unique graph of minimum degree 7, not 3. In this paper we obtain more precise information by determining the maximum number of s-cliques of a graph with prescribed order, circumference and minimum degree. As a corollary of our main result, we determine the maximum size of a k-connected graph with given order and circumference. Consequently we solve the corresponding problem for longest paths.

Motion Characteristics of Gas–Liquid Two-Phase Flow of Microbubbles in a Labyrinth Channel Used for Aerated Drip Irrigation

The indefinite characteristics of gas–liquid two-phase flow limit the usage of aerated drip irrigation. Gas–liquid two-phase flow in a labyrinth channel was observed using a particle tracking velocimetry (PTV) technique in this study. The motion trajectory and velocity vector of large numbers of microbubbles were characterized and analyzed at 0.01, 0.02, 0.04 MPa inlet pressure and in three labyrinth channels with different geometries. The results indicated that bubbly flow was the typical flow pattern in a labyrinth channel, with slug flow occurring occasionally. Smooth and gliding motion trajectories of bubbles were observed in the mainstream zone, while twisted trajectories were seen in the vortex zone. Increasing the inlet pressure increased the number of bubbles and the trajectory length in the vortex zone. When the inlet pressure increased from 0.02 to 0.04 MPa, the 25th percentile of Rc-t (the Ratio of Circular path length in the vortex zone to the Total trajectory length for a single bubble) increased from 0 to 12.3%, 0 to 6.1%, and 0 to 5.2% for channels A, B, and C, respectively; the 75th percentile increased from 31.3% to 43.9%, 27.5% to 31.9%, and 18.7% to 22.3%. The velocity vectors of the bubbles showed position dependence. Bubbles with high speed were found in the mainstream zone with their directions parallel to the water flow direction. Bubbles with low speed were seen in the vortex zone, moving in all directions. With inlet pressure increased from 0.01 to 0.04 MPa, the mean instantaneous velocities of bubbles in channels A, B, and C are increased by 106.2%, 107.6%, and 116.6%, respectively. At 0.04 MPa, channel A has the longest path length and the highest instantaneous velocity of bubbles in the vortex zone among three channels, exhibiting the highest anti-clogging performance of the three channels. This study will help in the comprehensive understanding of gas–liquid two-phase flow in a labyrinth channel used for aerated drip irrigation.

TB-RPL: A Try-the-Best Fused Mode of Operation to Enhance Point-to-Point Communication Performance in RPL

RPL is the IPv6 routing protocol for low-power and lossy networks in the Internet of Things which supports point-to-point (P2P) communication. However, the partition of two modes of operations (MOPs) in downward routing complicates achieving high performance. In the non-storing mode, a downward route with the longest path length is often picked. In the storing mode, the downward routes to some child nodes cannot be stored by their parent because of the limitation of memory space, which makes some nodes unreachable. In addition, there are extra performance costs of mixing or switching the two modes in the existing hybrid-MOPs works. Therefore, this article proposes TB-RPL to achieve an enhancement of RPL with a better performance of P2P communication. It allows all nodes to behave in a single and uniformly fused MOP that solves the problems mentioned above. The proposed mode uses a modified routing header format and introduces a threshold to the number of route entries. We implemented and compared TB-RPL with related mechanisms in Cooja simulator based on the Contiki-NG operating system. Simulation results verify that TB-RPL eliminates the three identified problems. Consequently, it significantly improves the performance of P2P communication in LLN.

The Longest Path Problem in Odd-Sized O-Shaped Grid Graphs

One of the well-known NP-hard optimization problems in graph theory is finding the longest path in a graph. This problem remains NP-hard for general grid graphs, and its complexity is open for grid graphs that have a limited number of holes. In this paper, we study this problem for odd-sized [Formula: see text]-shaped grid graphs, i.e. a rectangular grid graph with a rectangular hole. We show that this problem can be solved in linear time.

Energy Efficient Wireless Crowd Labeling: Joint Annotator Clustering and Power Control

The unprecedented growth of mobile data traffic has fueled the deployment of artificial intelligence (AI) at the network edge, while distilling the intelligence from raw data by machine learning requires tremendous labelling effort. To overcome this challenge, wireless crowd labelling (WCL) is proposed for efficient data labelling by exploiting billions of available mobile annotators and the multicasting property of wireless channels. A WCL system is considered in this paper where unlabelled data (objects) are multicast via fading channels to different clusters of annotators for repetition labelling to improve the accuracy. Given the desired labelling accuracy, the superposition coding technique together with the repetition labelling scheme give rise to a new tradeoff between radio-and-annotator resource consumption. Building on such tradeoff, the annotator clustering and transmit power control are jointly optimized to maximize the labelling throughput (i.e., the number of labelled objects) or minimize the power consumption, resulting in NP-hard integer programming problems. To solve these problems, the optimal structure of annotator clustering is derived by exploiting the property that the power allocation for multicasting objects tends to compensate for the worst channel among the annotators in each cluster. Based on such structure, the throughput maximization problem can be recognized as a longest-path problem and solved by means of branch-and-bound, while the power minimization problem can be recasted to a shortest-path problem and solved by means of forward dynamic programming. The solution approaches can be further simplified when the channels are symmetric by merging the same nodes and cutting the identical paths in the path graph. In addition, exact polices are derived for the special cases where either the annotators or power are constrained. Last, simulation results are presented to demonstrate the performance of our proposed joint designs.

All longest cycles in a 2‐connected partial 3‐tree share a common vertex

AbstractIn 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. In this paper, we study the related question on longest cycles instead of longest paths (to make the question valid, we require 2‐connectivity). The answer for cycles is also negative and not so much attention has been given to it. Classes for which the answer is known to be positive are dually chordal graphs and 3‐trees. Perhaps the main open class for the question of both paths and cycles is the class of chordal graphs. In this paper, we show that all longest cycles intersect in graphs with treewidth at most three. This class of graphs includes several important subclasses of graphs, such as series‐parallel graphs and ‐free chordal graphs.

SDSFLF: fault localization framework for optical communication using software digital switching network

Optical network is an emerging technology for data communication inworldwide. The information is transmitted from the source to destination through the fiber optics. All optical network (AON) provides good transmission transparency, good expandability, large bandwidth, lower bit error rate (BER), and high processing speed. Link failure and node failure haveconsistently occurred in the traditional methods. In order to overcome the above mentioned issues, this paper proposes a robust software defined switching enabled fault localization framework (SDSFLF) to monitor the node and link failure in an AON. In this work, a novel faulty node localization (FNL) algorithm is exploited to locate the faulty node. Then, the software defined faulty link detection (SDFLD) algorithm that addresses the problem of link failure. The failures are localized in multi traffic stream (MTS) and multi agent system (MAS). Thus, the throughput is improved in SDSFLF compared than other existing methods like traditional routing and wavelength assignment (RWA), simulated annealing (SA) algorithm, attackaware RWA (A-RWA) convex, longest path first (LPF) ordering, and biggest source-destination node degree (BND) ordering. The performance of the proposed algorithm is evaluated in terms of network load, wavelength utilization, packet loss rate, and burst loss rate. Hence, proposed SDSFLF assures that high performance is achieved than other traditional techniques.

METHOD OF ROUTING A GROUP OF MOBILE ROBOTS IN A FIXED NETWORK FOR SEARCHING THE MISSING OBJECTS IN A TECHNOLOGICAL DISASTER ZONE

Context. The relevance of the article is determined by the need for further development of models of collective behavior of systems with multi-agent structure construction endowed with intelligence that ensures synchronization of the joint efforts of various agents while achieving the goals set for the system. The method proposed in the article solves the problem of competition between different agents of a multi-agent system, which is important while performing search, rescue, and monitoring tasks in crisis areas of various origins.
 Objective is to develop a method for determining the sufficient population of a multi-agent system and the optimal routes of movement of its individual elements in a stationary network for the most complete examination of a technological disaster zone (any given zone based on a certain transport network).
 Method. We implemented the concept of a dynamic programming to search for all possible edge-simple longest paths connecting the directed subsets of vertices-sources and vertices-sinks in the structure of the model weighted directed graph. To this end, the modified Dijkstra method was applied. The modification comprises representing the weights of the arcs of the modeling directed graph with the negative values, which are further used in calculations according to the Dijkstra method. After finding the next edgesimple longest path, the arcs that make up it are fixed in the memory of the computer system (in the route plan) and removed from the graph structure, and the process is iteratively repeated. The search for paths takes place as long as the transitive closure between the vertices that are part of the specified subsets of source vertices and sink vertices is preserved. The developed method makes it possible to find such a set of traffic routes for the elements of the multi-agent system, which maximizes the area examined by them in a technological disaster zone (or the number of checked objects on the traffic routes) in one “wave” of the search and distributes the elements of a multi-agent system by routes that do not have common areas. A derivative of the application of the developed method is the determination of a sufficient population of a multi-agent system for effective search activities within the defined zone.
 Results. 1) A method of routing a group of mobile robots in a stationary network for searching the missing objects in a technological disaster zone has been developed. 2) The working expression of the Dijkstra method for searching in the structure of a network object (in the structure of a model graph) for the longest paths has been formalized. 3) We have suggested a set of indicators for a comprehensive evaluation of route plans of a multi-agent system. 4) The method has been verified on test problems.
 Conclusions. Theoretical studies and several experiments confirm the efficiency of the developed method. The solutions made using the developed method are accurate, which allows recommending it for practical use in determining in an automated mode route plans for multi-agent systems, as well as the required number of agents in such systems to perform the required amount of search tasks in a particular crisis area.

Non-Empty Intersection of Longest Paths in $H$-Free Graphs

We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $\kappa(G)$, and its independence number $\alpha(G)$ satisfies $\alpha(G) \le \kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.

Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices

The graph theory plays an important role in the network analysis, social networking as well as in many engineering fields such as electrical circuits, artificial intelligence, architecture, making the design or pattern of roads, buildings, shopping mall and etc. Due to this wide range application human enjoying her life with peacefully, Graph theory creates a way for human being to connect among themselves by social network. All above applications based on graph or molecule which may be the planer, non-planer and Peterson graph or etc. Peterson graph is the most important and reasonable example of Hypo-Hamiltonian graph. In the earlier, it was found as a hypo-traceable graph (graph which has not Hamiltonian graph. Naeem et al has worked on “A Two-Connected Graph with Gallai’s Property” In his research paper he has applied the property and has found the longest path and cycle in the graph. In this research paper we will develop the 3-dimensional graph of computational molecule contains 19 vertices and will split it into three different planes (xy, xz and yz-plane), and will find the longest path, longest cycle the molecule. The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.

Path Unselection for Path Delay Fault Test Generation

Path selection procedures identify path delay faults whose tests detect small delay defects. Path selection criteria are positive in the sense that they point to paths that should be selected, e.g., the longest paths. However, the longest paths are in many cases untestable, and the number of paths can be large when the longest testable paths are considered. For this scenario, it is advantageous to have a negative path selection procedure that excludes paths from consideration as targets for test generation. This provides the flexibility to identify detectable path delay faults among a reduced number of path delay faults. Such a procedure is developed in this article and referred to as a path unselection procedure. The procedure performs linear-time traversals of the circuit to unselect fan-out branches, and thus exclude from consideration paths that are shorter than the ones it keeps. It also uses available information about detectable and undetectable path delay faults to unselect fan-out branches or avoid unselecting them. To demonstrate its use, the procedure is applied to benchmark circuits as part of an iterative test generation procedure whose goal is to target manageable numbers of the longest path delay faults.

ARBic: an all-round biclustering algorithm for analyzing gene expression data.

Identifying significant biclusters of genes with specific expression patterns is an effective approach to reveal functionally correlated genes in gene expression data. However, none of existing algorithms can simultaneously identify both broader and narrower biclusters due to their failure of balancing between effectiveness and efficiency. We introduced ARBic, an algorithm which is capable of accurately identifying any significant biclusters of any shape, including broader, narrower and square, in any large scale gene expression dataset. ARBic was designed by integrating column-based and row-based strategies into a single biclustering procedure. The column-based strategy borrowed from RecBic, a recently published biclustering tool, extracts narrower biclusters, while the row-based strategy that iteratively finds the longest path in a specific directed graph, extracts broader ones. Being tested and compared to other seven salient biclustering algorithms on simulated datasets, ARBic achieves at least an average of 29% higher recovery, relevance and[Formula: see text] scores than the best existing tool. In addition, ARBic substantially outperforms all tools on real datasets and is more robust to noises, bicluster shapes and dataset types.

1930‒1990 - THE INTERVENTION OF THE BRAZILIAN STATE IN THE SUGAR-ENERGY SECTOR: IAA-PLANALSUCAR, PROÁLCOOL AND THE PARTICIPATION OF GILBERTO MILLER AZZI. DID WORK?

In Brazil, sugarcane is the crop whose agro-industrial development has taken the longest path and has the greatest prominence in the national agro-industry, without precedent. With 500 years of history, Brazil is today the largest producer of sugarcane raw material and sugarcane ethanol in the world. In addition, has the world's largest program for the replacement of fossil fuel by ethanol in vehicles and shares with India the position of largest sugar producer; in addition, it has developed the largest active sugarcane genetic improvement program in the world. However, in contrast to these achievements throughout its history, it has also received criticism even from businesspersons of the sector itself, mainly regarding State intervention and its inefficiency. This is a paradox: “success and inefficiency?” The general objective of this work was to analyze exactly this contradiction since the great agro-industrial development occurred especially during the 60 years of State intervention (1930‒1990). We concluded that this intervention was a key element in the success of the agroindustry both during and after the intervention. Many personalities deserve mention in this process, in both the field of ideas and decisions and practical actions: politicians, public agents, businesspersons, sugar mill owners, agroindustry technicians, and independent producers. We selected a researcher, Dr. Gilberto Miller Azzi, as an example, due to the fact that, from 1955 onwards, as a researcher at the IAA, he represented the scientific and technological area of the Brazilian State, acting decisively for the achievements of the sector. In this regard, an analysis was made of the main factors that have leveraged the national sugar and alcohol sector from the 1930s to the present day.

Prediction of influential proteins and enzymes of certain diseases using a directed unimodular hypergraph.

Protein-protein interaction (PPI) analysis based on mathematical modeling is an efficient means of identifying hub proteins, corresponding enzymes and many underlying structures. In this paper, a method for the analysis of PPI is introduced and used to analyze protein interactions of diseases such as Parkinson's, COVID-19 and diabetes melitus. A directed hypergraph is used to represent PPI interactions. A novel directed hypergraph depth-first search algorithm is introduced to find the longest paths. The minor hypergraph reduces the dimension of the directed hypergraph, representing the longest paths and results in the unimodular hypergraph. The property of unimodular hypergraph clusters influential proteins and enzymes that are related thereby providing potential avenues for disease treatment.

Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation

Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution $F$ supported on $[−\infty, 1$] with essential supremum equal to 1 (a charge of $−\infty$ is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by $C(F)$. Even in the simplest case where $F = p\delta_1 + (1 − p)\delta_{-\infty}$, corresponding to the longest path in the Barak-Erdős random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call Max Growth System (MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant $C(F)$. Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional.