Problem In Graph Theory Research Articles

Grafos hamiltonianos aplicado al turismo de Panamá

Un problema clásico de Teoría de Grafos es encontrar un camino que pase por varios puntos, sólo una vez, empezando y terminando en un lugar (camino hamiltoniano). Al agregar la condición de que sea la ruta más corta, el problema se convierte uno de tipo TSP (Traveling Salesman Problem). En este trabajo nos centraremos en un problema de tour turístico por la ciudad de Panamá, transformándolo a un problema de grafo de tal manera que represente la situación planteada.

Automatic Inference of Sound Correspondence Patterns across Multiple Languages

Sound correspondence patterns play a crucial role for linguistic reconstruction. Linguists use them to prove language relationship, to reconstruct proto-forms, and for classical phylogenetic reconstruction based on shared innovations. Cognate words that fail to conform with expected patterns can further point to various kinds of exceptions in sound change, such as analogy or assimilation of frequent words. Here I present an automatic method for the inference of sound correspondence patterns across multiple languages based on a network approach. The core idea is to represent all columns in aligned cognate sets as nodes in a network with edges representing the degree of compatibility between the nodes. The task of inferring all compatible correspondence sets can then be handled as the well-known minimum clique cover problem in graph theory, which essentially seeks to split the graph into the smallest number of cliques in which each node is represented by exactly one clique. The resulting partitions represent all correspondence patterns that can be inferred for a given data set. By excluding those patterns that occur in only a few cognate sets, the core of regularly recurring sound correspondences can be inferred. Based on this idea, the article presents a method for automatic correspondence pattern recognition, which is implemented as part of a Python library which supplements the article. To illustrate the usefulness of the method, I present how the inferred patterns can be used to predict words that have not been observed before.

Generalized Permanental Polynomials of Graphs

The search for complete graph invariants is an important problem in graph theory and computer science. Two networks with a different structure can be distinguished from each other by complete graph invariants. In order to find a complete graph invariant, we introduce the generalized permanental polynomials of graphs. Let G be a graph with adjacency matrix A ( G ) and degree matrix D ( G ) . The generalized permanental polynomial of G is defined by P G ( x , μ ) = per ( x I − ( A ( G ) − μ D ( G ) ) ) . In this paper, we compute the generalized permanental polynomials for all graphs on at most 10 vertices, and we count the numbers of such graphs for which there is another graph with the same generalized permanental polynomial. The present data show that the generalized permanental polynomial is quite efficient for distinguishing graphs. Furthermore, we can write P G ( x , μ ) in the coefficient form ∑ i = 0 n c μ i ( G ) x n − i and obtain the combinatorial expressions for the first five coefficients c μ i ( G ) ( i = 0 , 1 , ⋯ , 4 ) of P G ( x , μ ) .

Two-point resistances in a family of self-similar [formula omitted]-flower networks

The computation of resistance between two nodes in networks is a fundamental problem in both electric theory and graph theory. In this paper, first, a recursive algorithm for computing resistance between any two nodes in a family of self-similar (x,y)-flower networks is given. The (x,y)-flower networks display rich behavior as observed in a large variety of real systems. Then as explanations, using the algorithm, explicit expressions for some resistances in (1,3)-flower networks and (2,2)-flower networks are obtained.

Listing all maximal cliques in large graphs on vertex-centric model

Maximal Clique Enumeration (MCE), which consists to enumerate all maximal complete subgraphs in a given graph, is a fundamental problem in graph theory, and it is used in several applications. It is one of Karp’s 21 NP-complete problems. In the literature, this problem has been widely studied. One of the most notable, efficient, successful and extensively used solutions is the Bron–Kerbosch (BK) algorithm. The latter is a sequential algorithm which is able to enumerate all maximal cliques in an undirected graph without duplication. Furthermore, it is used to solve large problems such as maximum clique problem, community detection and graph clustering. However, for large graphs, sequential algorithms are slow and do not scale well. Thus, processing efficiently this kind of graphs needs to develop distributed algorithms under parallel and distributed platforms or large graph mining frameworks. In this setting, we propose new efficient distributed algorithms for maximal clique enumerating based on the vertex-centric model. These algorithms use the BK algorithm principle to deal with the MCE problem on large cluster. The proposed algorithms are implemented in Giraph and evaluated by using real-world graphs and computer-generated benchmark networks. Our experiments on a Hadoop cluster show that the proposed algorithms can effectively process a variety of large real-world and computer-generated graphs and scale well with increasing the dataset size and the number of nodes in the cluster. Furthermore, the proposed algorithms are provably work-efficient compared with other algorithms including BK algorithm.

A novel algorithm for the vertex cover problem based on minimal elements of discernibility matrix

Minimal vertex cover problem (MVCP) is a famous important NP-hard problem in graph theory. It has been reported that MVCP is equivalent to finding reducts of information systems in rough sets theory. This relationship motivates our idea to deal with MVCP in terms of approaches to discernibility matrix in rough set. First we point out that only minimal elements in the discernibility matrix are useful for MVCP, and we present a novel algorithm based on minimal elements for MVCP. Then we make experimental comparisons to demonstrate the effectiveness of this new algorithm on big graphs.

Scaffolding Based on Problems in Graph Theory Learning

Scaffolding is assistance provided so students can solve problems or understand concepts that can not be solved independently. When students are considered capable of completing their tasks independently, the assistance is eliminated. In this study, scaffolding was carried out by giving problems about graph theory. Problems given are closed and open contextual problems. The selection of this problem can be seen by students' understanding of graph theory. The results of this study, this problem-based scaffolding can help students who experience difficulties in the problem of graph theory. Students are enthusiastic because through this problem, they can explore and connect blood donation problems and regional tourism with the concept of graph.

Algorithm for the Vertex Connectivity Problem on Circular Trapezoid Graphs

The vertex connectivity k(G) of a graph G is the minimum number of nodes whose deletion disconnects it. Graph connectivity is one of the most fundamental problems in graph theory. In this paper, we designed an O(n2) time algorithm to solve connectivity problem on circular trapezoid graphs.

Price of Connectivity for the Vertex Cover Problem and the Dominating Set Problem: Conjectures and Investigation of Critical Graphs

The vertex cover problem and the dominating set problem are two well-known problems in graph theory. Their goal is to find the minimum size of a vertex subset satisfying some properties. Both hold a connected version, which imposes that the vertex subset must induce a connected component. To study the interdependence between the connected version and the original version of a problem, the Price of Connectivity ( $$PoC$$ ) was introduced by Cardinal and Levy (Theor Comput Sci 411(26–28):2581–2590, 2010) and Levy (Approximation algorithms for covering problems in dense graphs. Ph.D. thesis, Universite libre de Bruxelles, Brussels, 2009) as the ratio between invariants from the connected version and the original version of the problem. Camby et al. (Discret Math Theor Comput Sci 16:207–224, 2014) for the vertex cover problem, Camby and Schaudt (Discret Appl Math 177:53–59, 2014) for the dominating set problem characterized some classes of $$PoC$$ -Near-Perfect graphs, hereditary classes of graphs in which the Price of Connectivity is bounded by a fixed constant. Moreover, only for the vertex cover problem, Camby et al. (2014) introduced the notion of critical graphs, graphs that can appear in the list of forbidden induced subgraphs characterization. By definition, the Price of Connectivity of a critical graph is strictly greater than that of any proper induced subgraph. In this paper, we prove that for the vertex cover problem, every critical graph is either isomorphic to a cycle on 5 vertices or bipartite. To go further in the previous studies, we also present conjectures on $$PoC$$ -Near-Perfect graphs and critical graphs with the help of the computer software GraphsInGraphs (Camby and Caporossi in Studying graphs and their induced subgraphs with the computer: GraphsInGraphs. Cahiers du GERAD G-2016-10, 2016). Moreover, for the dominating set problem, we investigate critical trees and we show that every minimum dominating set of a critical graph is independent.

Social Structure Decomposition With Security Issue

Social structure decomposition has been a valuable research topic in the study of social networks, and it still has many unresolved problems. At the same time, an increasing number of people pays attention to the security issue that is worth considering and researching. With security consideration, we study a new social structure decomposition problem that can be formulated as a minimization problem in graph theory as follows: given a social network with formulation as a graph $G$ , partition the vertex set of $G$ into the minimum number of subsets, such that each subset induces an acyclic graph called a forest. This minimum number is called the vertex arboricity, denoted by $va(G)$ . It is well known that the vertex arboricity $va(G)\leq 3$ for each planar graph $G$ in the graph theory. In this paper, we prove that for the planar graph $G$ , if no 3-cycle intersects a 4-cycle or no 3-cycle intersects a 5-cycle, then $va(G)\leq 2$ .

On hamiltonian decompositions of tensor products of graphs

Finding a hamiltonian decomposition of G is one of the challenging problems in graph theory. We do not know for what classes of graphs G and H, their tensor product G x H is hamiltonian decomposable. In this paper, we have proved that, if G is a hamiltonian decomposable circulant graph with certain properties and H is a hamiltonian decomposable multigraph, then G x H is hamiltonian decomposable. In particular, tensor products of certain sparse hamiltonian decomposable circulant graphs are hamiltonian decomposable.

A Genetic Algorithm for the Snake-in-the-Box Problem using Modified Edge Recombination

Abstract Snake-in-the-box is a graph theory problem of finding the longest path in a hypercube with some constraints. As the dimension of the hypercube increases, the number of paths grow, increasing the search space exponentially. In the past, genetic algorithms have been known to work well with problems having such large search spaces. In this paper, a genetic algorithm using a modification of the edge recombination operator is presented to find the longest snake in dimensions one to nine with lesser edge failures.

Cost-Effective Edge Server Placement in Wireless Metropolitan Area Networks.

Remote clouds are gradually unable to achieve ultra-low latency to meet the requirements of mobile users because of the intolerable long distance between remote clouds and mobile users and the network congestion caused by the tremendous number of users. Mobile edge computing, a new paradigm, has been proposed to mitigate aforementioned effects. Existing studies mostly assume the edge servers have been deployed properly and they just pay attention to how to minimize the delay between edge servers and mobile users. In this paper, considering the practical environment, we investigate how to deploy edge servers effectively and economically in wireless metropolitan area networks. Thus, we address the problem of minimizing the number of edge servers while ensuring some QoS requirements. Aiming at more consistence with a generalized condition, we extend the definition of the dominating set, and transform the addressed problem into the minimum dominating set problem in graph theory. In addition, two conditions are considered for the capacities of edge servers: one is that the capacities of edge servers can be configured on demand, and the other is that all the edge servers have the same capacities. For the on-demand condition, a greedy based algorithm is proposed to find the solution, and the key idea is to iteratively choose nodes that can connect as many other nodes as possible under the delay, degree and cluster size constraints. Furthermore, a simulated annealing based approach is given for global optimization. For the second condition, a greedy based algorithm is also proposed to satisfy the capacity constraint of edge servers and minimize the number of edge servers simultaneously. The simulation results show that the proposed algorithms are feasible.

Joint Admission Control and Resource Allocation for Multi-Carrier Uplink NOMA Networks

In this letter, we study the maximum access problem (MAP) for uplink networks with non-orthogonal multiple access technique. Specifically, the MAP is formulated as a mixed integer programming problem with admission control, user clustering, channel assignment, and power control taken into account. To deal with this challenging problem, we first recast it as the maximum independent set (MIS) problem in graph theory. Then, we propose an efficient algorithm to solve this MIS problem in a low complexity manner. Finally, simulation results indicate that our algorithm can greatly improve the number of supported users compared to other schemes. The performance of our algorithm under different channel conditions and non-uniform distributions of users is also investigated.

Economic optimization in transient processes for model predictive control with a dynamic reference trajectory

In this paper, we focus on generating the reference trajectory to improve the economic performance in transient processes for model predictive control (MPC). We convert the trajectory generation problem to the shortest path problem which is one common problem in graph theory. For this purpose we construct a graph, also called the state-input graph, to approximate the dynamics of the system. The nodes of the graph are states obtained by mesh discretization of the state space and the weighted edges represent the costs when driving the system from one state to another under an admissible control input. The admissible control inputs are predetermined by mesh discretization of the input space. The graph describes the accessibility of the mesh points in the state space under control inputs in one sampling time. We run Dijkstra’s algorithm with the obtained graph to find a path from the current state to the optimal point at each sampling time. Then the path is sent to MPC as a reference. The procedure is repeated at the next sampling time and the reference is updated as well. Moreover, we extract a subgraph of the whole graph at each sampling time to accelerate the procedure. A numerical example and a benchmark chemical process are employed to demonstrate the effectiveness of the proposed approach.

Delay Analysis for End-to-End Synchronous Communication in Monitoring Systems.

With the rapid development of smart grid technologies, communication systems are further integrated in the existing power grids. The real-time capability and reliability of the power applications are receiving increasing concerns. Thus, it is important to measure the end-to-end delay in communication systems. The network calculus theory has been widely applied in the communication delay measuring tasks. However, for better operation performance of power systems, most power applications require synchronous data communication, in which the network calculus theory cannot be directly applied. In this paper, we expand the network calculus theory such that it can be used to analyze the communication delay for power applications in smart grids. The problem of communication delay calculation for the synchronization system is converted into a maximum path problem in graph theory. Finally, our theoretical results are compared with the experimental ones obtained with the network simulation software EstiNet. The simulation results verify the feasibility and effectiveness of the proposed method.

A GENETIC ALGORITHM FOR SOLVING FUZZY SHORTEST PATH PROBLEMS WITH INTERVAL TYPE-2 FUZZY ARC LENGTHS

Shortest path problem is one of the most fundamental and well-known optimization problems in graph theory due to its various real-world applications. Fuzzy set can manage the uncertainty, associated with the information of a problem, where conventional mathematical models may fail to reveal satisfactory result. In most cases, shortest path problem in fuzzy graph, called fuzzy shortest path problem, uses type-1 fuzzy set as arc length. The uncertainty associated with the linguistic description of information is not represented properly by type-1 fuzzy set due to inexactness of human perception in the evaluation of membership degrees having crisp values. An interval type-2 fuzzy set is able to tackle this type of uncertainty. In this paper, we have proposed an algorithmic approach based on genetic algorithm for finding shortest path from a source node to a destination node in a fuzzy graph with interval type-2 fuzzy arc lengths. We have designed a new crossover operator which does not need mutation operation. The purpose of mutation operation has been taken care by the proposed crossover operation. We have compared our algorithm with two other existing genetic algorithms for the fuzzy shortest path problem, where superiority of the proposed algorithm is shown. To the best of our knowledge, no algorithm based on genetic algorithm exists in the literature for fuzzy shortest path problem with interval type-2 fuzzy arc lengths. A numerical example is used to illustrate the effectiveness of the proposed approach.

New BSP/CGM algorithms for spanning trees

Computing a spanning tree (ST) and a minimum ST (MST) of a graph are fundamental problems in graph theory and arise as a subproblem in many applications. In this article, we propose parallel algorithms to these problems. One of the steps of previous parallel MST algorithms relies on the heavy use of parallel list ranking which, though efficient in theory, is very time-consuming in practice. Using a different approach with a graph decomposition, we devised new parallel algorithms that do not make use of the list ranking procedure. We proved that our algorithms are correct, and for a graph [Formula: see text], [Formula: see text], and [Formula: see text], the algorithms can be executed on a Bulk Synchronous Parallel/Coarse Grained Multicomputer (BSP/CGM) model using [Formula: see text] communications rounds with [Formula: see text] computation time for each round. To show that our algorithms have good performance on real parallel machines, we have implemented them on graphics processing unit. The obtained speedups are competitive and showed that the BSP/CGM model is suitable for designing general purpose parallel algorithms.

A graph-based active learning method for classification of remote sensing images

Remote sensing (RS) technology provides essential data for monitoring the Earth. To fully utilize the data, image classification is often needed to convert data to information. The success of image classification methods greatly depends on the quality and quantity of training samples. To effectively select more informative training samples, this paper proposes a new active learning (AL) technique for classification of remote sensing (RS) images based on graph theory. A new diversity criterion is proposed based on geometrical features of the support vector machines (SVM) outputs. The diversity selection procedure is converted to the densest k-subgraph [Formula: see text] maximization problem in graph theory. The [Formula: see text] maximization problem is solved by a greedy algorithm. The proposed technique is compared with competing methods adopted in RS community. Experimental tests are performed on very high resolution (VHR) multispectral and hyperspectral images. Experimental results demonstrate that the proposed technique leads to comparable or even better classification accuracies with respect to competing methods on the two datasets.

SPLMax: Exploiting the Simple Path Introduced Locality for Maximum Flow Acceleration

As a basic problem in graph theory, the maximum flow (max-flow) problem has important applications in networking and communication related areas. The simple path introduced locality is implicit in classic max-flow algorithms, i.e., only the vertices in simple paths between source and sink are involved in max-flow calculation. However, this kind of locality is completely ignored in existing acceleration methods, which leads to a lot of useless calculations and seriously degrades the acceleration effect. We propose simple-path locality based max-flow acceleration algorithm (SPLMax) to address the problem, where an overlay graph is built and used to accelerate calculation by only including necessary vertices. Random graph based simulations show that with SPLMax, at best only 0.001% vertices (i.e., 1/71193) in the graph need to be involved in max-flow calculation. For the comparison using real-world graphs, SPLMax has the minimal pre-processing time (at most 109 times faster than other methods) and minimal average max-flow computation time (at most 4.3 times faster than other methods).