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Journal articles on the topic 'Graph problems'

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Author: Grafiati

Published: 4 June 2021

Last updated: 30 January 2023

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1

Itzhakov, Avraham, and Michael Codish. "Incremental Symmetry Breaking Constraints for Graph Search Problems." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 02 (April 3, 2020): 1536–43. http://dx.doi.org/10.1609/aaai.v34i02.5513.

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This paper introduces incremental symmetry breaking constraints for graph search problems which are complete and compact. We show that these constraints can be computed incrementally: A symmetry breaking constraint for order n graphs can be extended to one for order n + 1 graphs. Moreover, these constraints induce a special property on their canonical solutions: An order n canonical graph contains a canonical subgraph on the first k vertices for every 1 ≤ k ≤ n. This facilitates a “generate and extend” paradigm for parallel graph search problem solving: To solve a graph search problem φ on order n graphs, first generate the canonical graphs of some order k < n. Then, compute canonical solutions for φ by extending, in parallel, each canonical order k graph together with suitable symmetry breaking constraints. The contribution is that the proposed symmetry breaking constraints enable us to extend the order k canonical graphs to order n canonical solutions. We demonstrate our approach through its application on two hard graph search problems.

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Wu, Y., P. Austrin, T. Pitassi, and D. Liu. "Inapproximability of Treewidth and Related Problems." Journal of Artificial Intelligence Research 49 (April 6, 2014): 569–600. http://dx.doi.org/10.1613/jair.4030.

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Graphical models, such as Bayesian Networks and Markov networks play an important role in artificial intelligence and machine learning. Inference is a central problem to be solved on these networks. This, and other problems on these graph models are often known to be hard to solve in general, but tractable on graphs with bounded Treewidth. Therefore, finding or approximating the Treewidth of a graph is a fundamental problem related to inference in graphical models. In this paper, we study the approximability of a number of graph problems: Treewidth and Pathwidth of graphs, Minimum Fill-In, One-Shot Black (and Black-White) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as Minimum Cut Linear Arrangement and Interval Graph Completion. We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are NP-hard to approximate to within any constant factor in polynomial time.

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Lynch, Mark A. M. "Creating recreational Hamiltonian cycle problems." Mathematical Gazette 88, no. 512 (July 2004): 215–18. http://dx.doi.org/10.1017/s0025557200174935.

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In this paper graphs that contain unique Hamiltonian cycles are introduced. The graphs are of arbitrary size and dense in the sense that their average vertex degree is greater than half the number of vertices that make up the graph. The graphs can be used to generate challenging puzzles. The problem is particularly challenging when the graph is large and the ‘method’ of solution is unknown to the solver.

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Dhamala, Tanka Nath. "Some Discrete Optimization Problems With Hamming and H-Comparability Graphs." Tribhuvan University Journal 27, no. 1-2 (December 30, 2010): 167–76. http://dx.doi.org/10.3126/tuj.v27i1-2.26400.

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Any H-comparability graph contains a Hamming graph as spanningsubgraph. An acyclic orientation of an H-comparability graph contains an acyclic orientation of the spanning Hamming graph, called sequence graph in the classical open-shop scheduling problem. We formulate different discrete optimization problems on the Hamming graphs and on H-comparability graphs and consider their complexity and relationship. Moreover, we explore the structures of these graphs in the class of irreducible sequences for the open shop problem in this paper.

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Sciriha, Irene. "Joining Forces for Reconstruction Inverse Problems." Symmetry 13, no. 9 (September 13, 2021): 1687. http://dx.doi.org/10.3390/sym13091687.

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A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.

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Golumbic, M. C., H. Kaplan, and R. Shamir. "Graph Sandwich Problems." Journal of Algorithms 19, no. 3 (November 1995): 449–73. http://dx.doi.org/10.1006/jagm.1995.1047.

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Makarov, Ilya, Dmitrii Kiselev, Nikita Nikitinsky, and Lovro Subelj. "Survey on graph embeddings and their applications to machine learning problems on graphs." PeerJ Computer Science 7 (February 4, 2021): e357. http://dx.doi.org/10.7717/peerj-cs.357.

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Dealing with relational data always required significant computational resources, domain expertise and task-dependent feature engineering to incorporate structural information into a predictive model. Nowadays, a family of automated graph feature engineering techniques has been proposed in different streams of literature. So-called graph embeddings provide a powerful tool to construct vectorized feature spaces for graphs and their components, such as nodes, edges and subgraphs under preserving inner graph properties. Using the constructed feature spaces, many machine learning problems on graphs can be solved via standard frameworks suitable for vectorized feature representation. Our survey aims to describe the core concepts of graph embeddings and provide several taxonomies for their description. First, we start with the methodological approach and extract three types of graph embedding models based on matrix factorization, random-walks and deep learning approaches. Next, we describe how different types of networks impact the ability of models to incorporate structural and attributed data into a unified embedding. Going further, we perform a thorough evaluation of graph embedding applications to machine learning problems on graphs, among which are node classification, link prediction, clustering, visualization, compression, and a family of the whole graph embedding algorithms suitable for graph classification, similarity and alignment problems. Finally, we overview the existing applications of graph embeddings to computer science domains, formulate open problems and provide experiment results, explaining how different networks properties result in graph embeddings quality in the four classic machine learning problems on graphs, such as node classification, link prediction, clustering and graph visualization. As a result, our survey covers a new rapidly growing field of network feature engineering, presents an in-depth analysis of models based on network types, and overviews a wide range of applications to machine learning problems on graphs.

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8

Halin, R. "Minimization Problems for Infinite n-Connected Graphs." Combinatorics, Probability and Computing 2, no. 4 (December 1993): 417–36. http://dx.doi.org/10.1017/s096354830000081x.

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A graph G is called n-minimizable if it can be reduced, by deleting a set of its edges, to a minimally n-connected graph. It is shown that, if n-connected graphs G and H differ only by finitely many vertices and edges, then G is n-minimizable if and only if H is n-minimizable (Theorem 4.12). In the main result, conditions are given that a tree decomposition of an n-connected graph G must satisfy in order to guarantee that the n-minimizability of each of the members of this decomposition implies the n-minimizability of the graph G (Theorem 6.5).

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Talebi, A. A., G. Muhiuddin, S. H. Sadati, and Hossein Rashmanlou. "New concepts of domination in fuzzy graph structures with application." Journal of Intelligent & Fuzzy Systems 42, no. 4 (March 4, 2022): 3705–18. http://dx.doi.org/10.3233/jifs-211923.

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Fuzzy graphs have a prominent place in the mathematical modelling of the problems due to the simplicity of representing the relationships between topics. Gradually, with the development of science and in encountering with complex problems and the existence of multiple relationships between variables, the need to consider fuzzy graphs with multiple relationships was felt. With the introduction of the graph structures, there was better flexibility than the graph in dealing with problems. By combining a graph structure with a fuzzy graph, a fuzzy graph structure was introduced that increased the decision-making power of complex problems based on uncertainties. The previous definitions restrictions in fuzzy graphs have made us present new definitions in the fuzzy graph structure. The domination of fuzzy graphs has many applications in other sciences including computer science, intelligent systems, psychology, and medical sciences. Hence, in this paper, first we study the dominating set in a fuzzy graph structure from the perspective of the domination number of its fuzzy relationships. Likewise, we determine the domination in terms of neighborhood, degree, and capacity of vertices with some examples. Finally, applications of domination are introduced in fuzzy graph structure.

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Makhnev, A. A., I. N. Belousov, and D. V. Paduchikh. "Inverse problems of graph theory: Graphs without triangles." Sibirskie Elektronnye Matematicheskie Izvestiya 18, no. 1 (January 21, 2021): 27–42. http://dx.doi.org/10.33048/semi.2021.18.003.

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Korpelainen, Nicholas, Vadim V. Lozin, Dmitriy S. Malyshev, and Alexander Tiskin. "Boundary properties of graphs for algorithmic graph problems." Theoretical Computer Science 412, no. 29 (July 2011): 3545–54. http://dx.doi.org/10.1016/j.tcs.2011.03.001.

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12

Wanke, Egon. "Algorithms for graph problems on BNLC structured graphs." Information and Computation 94, no. 1 (September 1991): 93–122. http://dx.doi.org/10.1016/0890-5401(91)90035-z.

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Prabha, S. Celine, M. Palanivel, S. Amutha, N. Anbazhagan, Woong Cho, Hyoung-Kyu Song, Gyanendra Prasad Joshi, and Hyeonjoon Moon. "Solutions of Detour Distance Graph Equations." Sensors 22, no. 21 (November 2, 2022): 8440. http://dx.doi.org/10.3390/s22218440.

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Graph theory is a useful mathematical structure used to model pairwise relations between sensor nodes in wireless sensor networks. Graph equations are nothing but equations in which the unknown factors are graphs. Many problems and results in graph theory can be formulated in terms of graph equations. In this paper, we solved some graph equations of detour two-distance graphs, detour three-distance graphs, detour antipodal graphs involving with the line graphs.

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Jiang, Huiqin, Ali Asghar Talebi, Zehui Shao, Seyed Hossein Sadati, and Hossein Rashmanlou. "New Concepts of Vertex Covering in Cubic Graphs with Its Applications." Mathematics 10, no. 3 (January 19, 2022): 307. http://dx.doi.org/10.3390/math10030307.

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Graphs serve as one of the main tools for the mathematical modeling of various human problems. Fuzzy graphs have the ability to solve uncertain and ambiguous problems. The cubic graph, which has recently gained a position in the fuzzy graph family, has shown good capabilities when faced with problems that cannot be expressed by fuzzy graphs and interval-valued fuzzy graphs. Simultaneous application of fuzzy and interval-valued fuzzy membership indicates a high flexibility in modeling uncertainty issues. The vertex cover is a fundamental issue in graph theory that has wide application in the real world. The previous definition limitations in the vertex covering of fuzzy graphs has directed us to offer new classifications in terms of cubic graph. In this study, we introduced the strong vertex covering and independent vertex covering in a cubic graph with strong edges and described some of its properties. One of the motives of this research was to examine the changes in the strong vertex covering number of a cubic graph if one vertex is omitted. This issue can play a decisive role in covering the graph vertices. Since many of the problems ahead are of hybrid type, by reviewing some operations on the cubic graph we were able to determine the strong vertex covering number on the most important cubic product operations. Finally, two applications of strong vertex covering and strong vertex independence are presented.

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15

Rahman, Saifur. "Semirings of graphs: homomorphisms and applications in network problems." Proyecciones (Antofagasta) 41, no. 6 (October 28, 2022): 1273–96. http://dx.doi.org/10.22199/issn.0717-6279-5137.

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This paper deals with studying some algebraic structures of the graphs as an attempt to visualize abstract mathematics. We have used some binary graph operations to investigated the algebraic structures of graphs with examples. This work emphasizes specifically the construction of semigroup or monoid and semiring, and their properties. This manuscript also aims to give a focused introduction of a class of homomorphism on the semiring of graphs. Some instances of real-life decision problems are consequently discussed. This article is also in a nascent stage of relating number theory and graph theory through mappings.

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Dhulipala, Laxman, Guy E. Blelloch, and Julian Shun. "Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable." ACM Transactions on Parallel Computing 8, no. 1 (April 2021): 1–70. http://dx.doi.org/10.1145/3434393.

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There has been significant recent interest in parallel graph processing due to the need to quickly analyze the large graphs available today. Many graph codes have been designed for distributed memory or external memory. However, today even the largest publicly-available real-world graph (the Hyperlink Web graph with over 3.5 billion vertices and 128 billion edges) can fit in the memory of a single commodity multicore server. Nevertheless, most experimental work in the literature report results on much smaller graphs, and the ones for the Hyperlink graph use distributed or external memory. Therefore, it is natural to ask whether we can efficiently solve a broad class of graph problems on this graph in memory. This paper shows that theoretically-efficient parallel graph algorithms can scale to the largest publicly-available graphs using a single machine with a terabyte of RAM, processing them in minutes. We give implementations of theoretically-efficient parallel algorithms for 20 important graph problems. We also present the interfaces, optimizations, and graph processing techniques that we used in our implementations, which were crucial in enabling us to process these large graphs quickly. We show that the running times of our implementations outperform existing state-of-the-art implementations on the largest real-world graphs. For many of the problems that we consider, this is the first time they have been solved on graphs at this scale. We have made the implementations developed in this work publicly-available as the Graph Based Benchmark Suite (GBBS).

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Bouyukliev, Iliya, and Mariya Dzhumalieva-Stoeva. "Representing Equivalence Problems for Combinatorial Objects." Serdica Journal of Computing 8, no. 4 (October 2, 2015): 327–54. http://dx.doi.org/10.55630/sjc.2014.8.327-354.

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Methods for representing equivalence problems of various combinatorial objectsas graphs or binary matrices are considered. Such representations can be used for isomorphism testing in classification or generation algorithms. Often it is easier to consider a graph or a binary matrix isomorphism problem than to implement heavy algorithms depending especially on particular combinatorialobjects. Moreover, there already exist well tested algorithms for the graph isomorphismproblem (nauty) and the binary matrix isomorphism problem as well (Q-Extension).ACM Computing Classification System (1998): F.2.1, G.4.

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Huang, Liangsong, Yu Hu, Yuxia Li, P. K. Kishore Kumar, Dipak Koley, and Arindam Dey. "A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications." Mathematics 7, no. 6 (June 17, 2019): 551. http://dx.doi.org/10.3390/math7060551.

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Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the d m - and t d m -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely d m -regular, t d m -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the μ -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here.

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19

Wang, Shiping, Qingxin Zhu, William Zhu, and Fan Min. "Equivalent Characterizations of Some Graph Problems by Covering-Based Rough Sets." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/519173.

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Covering is a widely used form of data structures. Covering-based rough set theory provides a systematic approach to this data. In this paper, graphs are connected with covering-based rough sets. Specifically, we convert some important concepts in graph theory including vertex covers, independent sets, edge covers, and matchings to ones in covering-based rough sets. At the same time, corresponding problems in graphs are also transformed into ones in covering-based rough sets. For example, finding a minimal edge cover of a graph is translated into finding a minimal general reduct of a covering. The main contributions of this paper are threefold. First, any graph is converted to a covering. Two graphs induce the same covering if and only if they are isomorphic. Second, some new concepts are defined in covering-based rough sets to correspond with ones in graph theory. The upper approximation number is essential to describe these concepts. Finally, from a new viewpoint of covering-based rough sets, the general reduct is defined, and its equivalent characterization for the edge cover is presented. These results show the potential for the connection between covering-based rough sets and graphs.

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Chen, Jong-Shin, Ruo-Wei Hung, Fatemeh Keshavarz-Kohjerdi, and Yung-Fa Huang. "Domination and Independent Domination in Extended Supergrid Graphs." Algorithms 15, no. 11 (October 30, 2022): 402. http://dx.doi.org/10.3390/a15110402.

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Supergrid graphs are derived by computing stitch paths for computerized embroidery machines. In the past, we have studied the Hamiltonian-related properties of supergrid graphs and their subclasses of graphs. In this paper, we propose a generalized graph class for supergrid graphs called extended supergrid graphs. Extended supergrid graphs include grid graphs, supergrid graphs, diagonal supergrid graphs, and triangular supergrid graphs as subclasses of graphs. In this paper, we study the problems of domination and independent domination on extended supergrid graphs. A dominating set of a graph is the subset of vertices on it, such that every vertex of the graph is in this set or adjacent to at least a vertex of this set. If any two vertices in a dominating set are not adjacent, this is called an independent dominating set. Domination and independent domination problems find a dominating set and an independent dominating set with the least number of vertices on a graph, respectively. The domination and independent domination set problems on grid graphs are known to be NP-complete, meaning that these two problems on extended supergrid graphs are also NP-complete. However, the complexities of these two problems in other subclasses of graphs remain unknown. In this paper, we first prove that these two problems on diagonal supergrid graphs are NP-complete, then, by a simple extension, we prove that these two problems on supergrid graphs and triangular supergrid graphs are also NP-complete. In addition, these two problems on rectangular supergrid graphs are known to be linearly solvable; however, the complexities of these two problems on rectangular triangular-supergrid graphs remain unknown. This paper provides tight upper bounds on the sizes of the minimum dominating and independent dominating sets for rectangular triangular-supergrid graphs.

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Király, Zoltán, and Jácint Szabó. "Induced Graph Packing Problems." Graphs and Combinatorics 26, no. 2 (February 27, 2010): 243–57. http://dx.doi.org/10.1007/s00373-010-0906-0.

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22

Marx, Dániel. "Parameterized graph separation problems." Theoretical Computer Science 351, no. 3 (February 2006): 394–406. http://dx.doi.org/10.1016/j.tcs.2005.10.007.

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23

Shamir, Ron, Roded Sharan, and Dekel Tsur. "Cluster graph modification problems." Discrete Applied Mathematics 144, no. 1-2 (November 2004): 173–82. http://dx.doi.org/10.1016/j.dam.2004.01.007.

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Marx, Dániel, and Ildikó Schlotter. "Parameterized graph cleaning problems." Discrete Applied Mathematics 157, no. 15 (August 2009): 3258–67. http://dx.doi.org/10.1016/j.dam.2009.06.022.

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25

Fahrenthold, E. P., and J. D. Wargo. "Lagrangian Bond Graphs for Solid Continuum Dynamics Modeling." Journal of Dynamic Systems, Measurement, and Control 116, no. 2 (June 1, 1994): 178–92. http://dx.doi.org/10.1115/1.2899209.

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The limitations of existing continuum bond graph modeling techniques have effectively precluded their use in large order problems, where nonrepetitive graph structures and causal patterns are normally present. As a result, despite extensive publication of bond graph models for continuous systems simulations, bond graph methods have not offered a viable alternative to finite element analysis for the vast majority of practical problems. However, a new modeling approach combining Lagrangian (mass fixed) bond graphs with a selected finite element discretization scheme allows for direct simulation of a wide range of large order solid continuum dynamics problems. With appropriate modifications, including the use of Eulerian (space fixed) bond graphs, the method may be extended to include fluid dynamics modeling.

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Chen, Lin, Li Zeng, Jin Peng, Junren Ming, and Xianghui Zhu. "Regularity Index of Uncertain Random Graph." Symmetry 15, no. 1 (January 3, 2023): 137. http://dx.doi.org/10.3390/sym15010137.

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A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry.

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DESMEDT, YVO, and YONGGE WANG. "ANALYZING VULNERABILITIES OF CRITICAL INFRASTRUCTURES USING FLOWS AND CRITICAL VERTICES IN AND/OR GRAPHS." International Journal of Foundations of Computer Science 15, no. 01 (February 2004): 107–25. http://dx.doi.org/10.1142/s0129054104002339.

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AND/OR graphs and minimum-cost solution graphs have been studied extensively in artificial intelligence (see, e.g., Nilsson [14]). Generally, the AND/OR graphs are used to model problem solving processes. The minimum-cost solution graph can be used to attack the problem with the least resource. However, in many cases we want to solve the problem within the shortest time period and we assume that we have as many concurrent resources as we need to run all concurrent processes. In this paper, we will study this problem and present an algorithm for finding the minimum-time-cost solution graph in an AND/OR graph. We will also study the following problems which often appear in industry when using AND/OR graphs to model manufacturing processes or to model problem solving processes: finding maximum (additive and non-additive) flows and critical vertices in an AND/OR graph. A detailed study of these problems provide insight into the vulnerability of complex systems such as cyber-infrastructures and energy infrastructures (these infrastructures could be modeled with AND/OR graphs). For an infrastructure modeled by an AND/OR graph, the protection of critical vertices should have highest priority since terrorists could defeat the whole infrastructure with the least effort by destroying these critical points. Though there are well known polynomial time algorithms for the corresponding problems in the traditional graph theory, we will show that generally it is NP-hard to find a non-additive maximum flow in an AND/OR graph, and it is both NP-hard and coNP-hard to find a set of critical vertices in an AND/OR graph. We will also present a polynomial time algorithm for finding a maximum additive flow in an AND/OR graph, and discuss the relative complexity of these problems.

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Hundack, Christoph, Hans Jürgen Prömel, and Angelika Steger. "Extremal Graph Problems for Graphs with a Color-Critical Vertex." Combinatorics, Probability and Computing 2, no. 4 (December 1993): 465–77. http://dx.doi.org/10.1017/s0963548300000833.

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In this paper we consider the following problem, given a graph H, what is the structure of a typical, i.e. random, H-free graph? We completely solve this problem for all graphs H containing a critical vertex. While this result subsumes a sequence of known results, its short proof is self contained.

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Alekseev, V. E., R. Boliac, D. V. Korobitsyn, and V. V. Lozin. "NP-hard graph problems and boundary classes of graphs." Theoretical Computer Science 389, no. 1-2 (December 2007): 219–36. http://dx.doi.org/10.1016/j.tcs.2007.09.013.

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Matsushita, Takahiro. "Answers to some problems about graph coloring test graphs." European Journal of Combinatorics 45 (April 2015): 59–64. http://dx.doi.org/10.1016/j.ejc.2014.10.006.

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LIN, YAW-LING, and STEVEN S. SKIENA. "COMPLEXITY ASPECTS OF VISIBILITY GRAPHS." International Journal of Computational Geometry & Applications 05, no. 03 (September 1995): 289–312. http://dx.doi.org/10.1142/s0218195995000179.

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In this paper, we consider two distinct problems related to complexity aspects of the visibility graphs of simple polygons. Recognizing visibility graphs is a long-standing open problem. It is not even known whether visibility graph recognition is in NP. That visibility graph recognition is in NP would be established if we could demonstrate that any n vertex visibility graph is realized by a polygon which can be drawn on an exponentially-sized grid. This motivates a study of the area requirements for realizing visibility graphs. In this paper, we prove: • Θ(n3) area is necessary and sufficient to realize the complete visibility graph Kn. • There exist visibility graphs which require exponential area to realize. • Any maximal outerplanar graph of diameter d can be realized in O(d2 · 2d) area, which can be as small as O(n log2 n) for a balanced mop. Linear maximal outer-planar graphs can be realized in O(n8) area. The second part of this paper considers the complexity of specific optimization problems on visibility graphs. Given a polygon P, we show that finding a maximum independent set, minimum vertex cover, or maximum dominating set in the visibility graph of P are all NP-complete. Further we show that for polygons P1 and P2, the problem of testing if they have isomorphic visibility graphs is isomorphism-complete. These problems remain hard when given the visibility graphs as input.

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Nazeer, Saqib, Muhammad Hussain, Fatimah Abdulrahman Alrawajeh, and Sultan Almotairi. "Metric Dimension on Path-Related Graphs." Mathematical Problems in Engineering 2021 (October 29, 2021): 1–12. http://dx.doi.org/10.1155/2021/2085778.

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Graph theory has a large number of applications in the fields of computer networking, robotics, Loran or sonar models, medical networks, electrical networking, facility location problems, navigation problems etc. It also plays an important role in studying the properties of chemical structures. In the field of telecommunication networks such as CCTV cameras, fiber optics, and cable networking, the metric dimension has a vital role. Metric dimension can help us in minimizing cost, labour, and time in the above discussed networks and in making them more efficient. Resolvability also has applications in tricky games, processing of maps or images, pattern recognitions, and robot navigation. We defined some new graphs and named them s − middle graphs, s -total graphs, symmetrical planar pyramid graph, reflection symmetrical planar pyramid graph, middle tower path graph, and reflection middle tower path graph. In the recent study, metric dimension of these path-related graphs is computed.

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Peng, Yun, Byron Choi, and Jianliang Xu. "Graph Learning for Combinatorial Optimization: A Survey of State-of-the-Art." Data Science and Engineering 6, no. 2 (April 28, 2021): 119–41. http://dx.doi.org/10.1007/s41019-021-00155-3.

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AbstractGraphs have been widely used to represent complex data in many applications, such as e-commerce, social networks, and bioinformatics. Efficient and effective analysis of graph data is important for graph-based applications. However, most graph analysis tasks are combinatorial optimization (CO) problems, which are NP-hard. Recent studies have focused a lot on the potential of using machine learning (ML) to solve graph-based CO problems. Most recent methods follow the two-stage framework. The first stage is graph representation learning, which embeds the graphs into low-dimension vectors. The second stage uses machine learning to solve the CO problems using the embeddings of the graphs learned in the first stage. The works for the first stage can be classified into two categories, graph embedding methods and end-to-end learning methods. For graph embedding methods, the learning of the the embeddings of the graphs has its own objective, which may not rely on the CO problems to be solved. The CO problems are solved by independent downstream tasks. For end-to-end learning methods, the learning of the embeddings of the graphs does not have its own objective and is an intermediate step of the learning procedure of solving the CO problems. The works for the second stage can also be classified into two categories, non-autoregressive methods and autoregressive methods. Non-autoregressive methods predict a solution for a CO problem in one shot. A non-autoregressive method predicts a matrix that denotes the probability of each node/edge being a part of a solution of the CO problem. The solution can be computed from the matrix using search heuristics such as beam search. Autoregressive methods iteratively extend a partial solution step by step. At each step, an autoregressive method predicts a node/edge conditioned to current partial solution, which is used to its extension. In this survey, we provide a thorough overview of recent studies of the graph learning-based CO methods. The survey ends with several remarks on future research directions.

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Noppakaew, Passawan, Kaewkan Hengthonglert, and Sawanya Sakuntasathien. "Dominating Broadcasts in Fuzzy Graphs." Mathematics 10, no. 2 (January 17, 2022): 281. http://dx.doi.org/10.3390/math10020281.

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Broadcasting problems in graph theory play a significant role in solving many complicated physical problems. However, in real life there are many vague situations that sometimes cannot be modeled using usual graphs. Consequently, the concept of a fuzzy graph GF:(V,σ,μ) has been introduced to deal with such problems. In this study, we are interested in defining the notion of dominating broadcasts in fuzzy graphs. We also show that, in a connected fuzzy graph containing more than one element in σ*, a dominating broadcast always exists, where σ* is {v∈V|σ(v)>0}. In addition, we investigate the relationship between broadcast domination numbers, radii, and domination numbers in a fuzzy graph as follows; γb(GF)≤min{r(GF),γ(GF)}, where γb(GF) is the broadcast domination number, r(GF) is the radius, and γ(GF) is domination numbers in fuzzy graph GF, with |σ*|>1.

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Wu, Yalong, and Haizhen Ren. "Complexity of the Operation Graphs on Cycle and Complete Graph." Journal of Physics: Conference Series 2333, no. 1 (August 1, 2022): 012006. http://dx.doi.org/10.1088/1742-6596/2333/1/012006.

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Abstract The number of spanning trees (or complexity) of a graph can be used to measure the reliabilities of network and circuit design. Although many NP hard problems are related to the complexity of graphs, there are explicit formulas for the complexity of some specific graph classes. This paper mainly studies the explicit expression of the complexity of graphs obtained by graph operations. Through classical graph operations, we obtain some operation graphs generated by cycle and complete graph, and get the closed formulas for the complexity in these operation graphs. Compared with Daoud’s results, we obtain some interesting algebraic properties about the complexity of operation graphs.

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Hasegawa, Ryoichi, and Hisashi Handa. "Solving Order/Degree Problems by Using EDA-GK with a Novel Sampling Method." Journal of Advanced Computational Intelligence and Intelligent Informatics 22, no. 2 (March 20, 2018): 236–41. http://dx.doi.org/10.20965/jaciii.2018.p0236.

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The Estimation of Distribution Algorithms with Graph Kernels called EDA-GK is an extension of the Estimation of Distribution Algorithms that can work with graph-related problems. Individuals of the EDA-GK are represented by graphs. In this paper, the EDA-GK is applied to solve for the Order/Degree problems, which are an NP-hard problems and are a benchmark problem in graph theory studies. Moreover, we incorporate a new sampling method for generating offspring. Experimental results on several problem instances of Order/Degree problems show the effectiveness of the EDA-GK.

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KURASOV, P. "Inverse problems for Aharonov–Bohm rings." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 2 (November 26, 2009): 331–62. http://dx.doi.org/10.1017/s030500410999034x.

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AbstractThe inverse problem for Schrödinger operators on metric graphs is investigated in the presence of magnetic field. Graphs without loops and with Euler characteristic zero are considered. It is shown that the knowledge of the Titchmarsh–Weyl matrix function (Dirichlet-to-Neumann map) for just two values of the magnetic field allows one to reconstruct the graph and potential on it provided a certain additional no-resonance condition is satisfied.

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38

Cvetkovic, Dragos. "Spectral recognition of graphs." Yugoslav Journal of Operations Research 22, no. 2 (2012): 145–61. http://dx.doi.org/10.2298/yjor120925025c.

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At some time, in the childhood of spectral graph theory, it was conjectured that non-isomorphic graphs have different spectra, i.e. that graphs are characterized by their spectra. Very quickly this conjecture was refuted and numerous examples and families of non-isomorphic graphs with the same spectrum (cospectral graphs) were found. Still some graphs are characterized by their spectra and several mathematical papers are devoted to this topic. In applications to computer sciences, spectral graph theory is considered as very strong. The benefit of using graph spectra in treating graphs is that eigenvalues and eigenvectors of several graph matrices can be quickly computed. Spectral graph parameters contain a lot of information on the graph structure (both global and local) including some information on graph parameters that, in general, are computed by exponential algorithms. Moreover, in some applications in data mining, graph spectra are used to encode graphs themselves. The Euclidean distance between the eigenvalue sequences of two graphs on the same number of vertices is called the spectral distance of graphs. Some other spectral distances (also based on various graph matrices) have been considered as well. Two graphs are considered as similar if their spectral distance is small. If two graphs are at zero distance, they are cospectral. In this sense, cospectral graphs are similar. Other spectrally based measures of similarity between networks (not necessarily having the same number of vertices) have been used in Internet topology analysis, and in other areas. The notion of spectral distance enables the design of various meta-heuristic (e.g., tabu search, variable neighbourhood search) algorithms for constructing graphs with a given spectrum (spectral graph reconstruction). Several spectrally based pattern recognition problems appear in many areas (e.g., image segmentation in computer vision, alignment of protein-protein interaction networks in bio-informatics, recognizing hard instances for combinatorial optimization problems such as the travelling salesman problem). We give a survey of such and other graph spectral recognition techniques used in computer sciences.

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Gurski, Frank, Dominique Komander, and Carolin Rehs. "Oriented Coloring on Recursively Defined Digraphs." Algorithms 12, no. 4 (April 25, 2019): 87. http://dx.doi.org/10.3390/a12040087.

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Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G = ( V , A ) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows that finding the chromatic number of an oriented graph is an NP-hard problem, too. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.

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LEVIN, LEONID A., and RAMARATHNAM VENKATESAN. "An Average Case NP-complete Graph Colouring Problem." Combinatorics, Probability and Computing 27, no. 5 (April 2, 2018): 808–28. http://dx.doi.org/10.1017/s0963548318000123.

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NP-complete problems should be hard on some instances but those may be extremely rare. On generic instances many such problems, especially related to random graphs, have been proved to be easy. We show the intractability of random instances of a graph colouring problem: this graph problem is hard on average unless all NP problems under all samplable (i.e. generatable in polynomial time) distributions are easy. Worst case reductions use special gadgets and typically map instances into a negligible fraction of possible outputs. Ours must output nearly random graphs and avoid any super-polynomial distortion of probabilities. This poses significant technical difficulties.

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Turab, Ali, Wutiphol Sintunavarat, and Jong-Suk Ro. "On Novel Mathematical Modeling for Studying a Class of Nonlinear Caputo-Type Fractional-Order Boundary Value Problems Emerging in CGT." Fractal and Fractional 7, no. 2 (January 17, 2023): 99. http://dx.doi.org/10.3390/fractalfract7020099.

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Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. Star graphs were used since their method requires a central point linked to other vertices but not to itself. Our objective is to expand the mechanism by introducing the idea of an isobutane graph that has the chemical formula C4H10 and CAS number 75-28-5. By using the appropriate fixed point theory findings, this paper investigates the existence of solutions to fractional boundary value problems of Caputo type on such graphs. Additionally, two examples are provided to strengthen our important conclusions.

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Caro, Y., and Y. Roditty. "On zero-sum turan problems of Bialostocki and Dierker." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 53, no. 3 (December 1992): 402–7. http://dx.doi.org/10.1017/s1446788700036569.

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AbstractAssume G is a graph with m edges. By T(n, G) we denote the classical Turan number, namely, the maximum possible number of edges in a graph H on n vertices without a copy of G. Similarly if G is a family of graphs then H does not have a copy of any member of the family. A Zk-colouring of a graph G is a colouring of the edges of G by Zk, the additive group of integers modulo k, avoiding a copy of a given graph H, for which the sum of the values on its edges is 0 (mod k). By the Zero-Sum Turan number, denoted T(n, G, Zk), k¦m, we mean the maximum number of edges in a Zk-colouring of a graph on n vertices that contains no zero-sum (mod k) copy of G. Here we mainly solve two problems of Bialostocki and Dierker [6].Problem 1. Determine T(n, tK2, Zk) for ¦|t. In particular, is it true that T(n, tK2, Zk) = T(n, (t+k-1)K2)?Problem 2. Does there exist a constant c(t, k) such that T(n, Ft, Zk) ≦ c(t, k)n, where Ft is the family of cycles of length at least t?

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Kusper, Gábor, and Csaba Biró. "Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model." Algorithms 13, no. 12 (December 3, 2020): 321. http://dx.doi.org/10.3390/a13120321.

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In a previous paper we defined the black and white SAT problem which has exactly two solutions, where each variable is either true or false. We showed that black and white 2-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonboglár model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a black and white SAT problem. We prove a powerful theorem, the so called transitions theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as black and white SAT problems. We show that the Balatonboglár model is between the strong and the weak model, and it generates 3-SAT problems, so the Balatonboglár model represents strongly connected communication graphs as black and white 3-SAT problems. Our motivation to study these models is the following: The strong model generates a 2-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonboglár model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonboglár model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: the SAT problem and directed graphs.

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Ye, Tian-Rui, and Haiying Wang. "Open Problems on Integral Sum Labellings." International Journal of Applied Mathematics and Machine Learning 13, no. 1 (September 20, 2020): 1–4. http://dx.doi.org/10.18642/ijamml_7100122153.

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Let be a simple graph and be the set of all integers. An integral sum graph of a set of integers as the graph having as its vertex set, with two vertices adjacent whenever their sum is in A graph so obtained is called an integral sum graph. In other words, an integral sum graph of a finite subset is the graph with if and only if And is called an integral sum labelling of In the paper, many obtained conclusions are summarized and relevant open problem is raised.

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Khan, Sami Ullah, Abdul Nasir, Naeem Jan, and Zhen-Hua Ma. "Graphical Analysis of Covering and Paired Domination in the Environment of Neutrosophic Information." Mathematical Problems in Engineering 2021 (April 17, 2021): 1–12. http://dx.doi.org/10.1155/2021/5518295.

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Neutrosophic graph (NG) is a powerful tool in graph theory, which is capable of modeling many real-life problems with uncertainty due to unclear, varying, and indeterminate information. Meanwhile, the fuzzy graphs (FGs) and intuitionistic fuzzy graphs (IFGs) may not handle these problems as efficiently as NGs. It is difficult to model uncertainty due to imprecise information and vagueness in real-world scenarios. Many real-life optimization problems are modeled and solved using the well-known fuzzy graph theory. The concepts of covering, matching, and paired domination play a major role in theoretical and applied neutrosophic environments of graph theory. Henceforth, the current study covers this void by introducing the notions of covering, matching, and paired domination in single-valued neutrosophic graph (SVNG) using the strong edges. Also, many attention-grabbing properties of these concepts are studied. Moreover, the strong covering number, strong matching number, and the strong paired domination number of complete SVNG, complete single-valued neutrosophic cycle (SVNC), and complete bipartite SVNG are worked out along with their fascinating properties.

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Luo, Haichang, Sakander Hayat, Yubin Zhong, Zhongyuan Peng, and Tamás Réti. "The IRC Indices of Transformation and Derived Graphs." Mathematics 10, no. 7 (March 30, 2022): 1111. http://dx.doi.org/10.3390/math10071111.

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An irregularity index IR(Γ) of a graph Γ is a nonnegative numeric quantity (i.e., IR(Γ)≥0) such that IR(Γ)=0 iff Γ is a regular graph. In this paper, we show that IRC closely correlates with the normal boiling point Tbp and the standard heat of formation ΔHfo of lower benzenoid hydrocarbons. The correlation models that fit the data efficiently for both Tbp and ΔHfo are linear. We develop further mathematical properties of IRC by calculating its exact expressions for the recently introduced transformation graphs as well as certain derived graphs, such as the total graph, semi-total point graph, subdivision graph, semi-total line graph, double, strong double, and extended double cover graphs. Some open problems are proposed for further research on the IRC index of graphs.

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Hanif, Muhammad Zeeshan, Naveed Yaqoob, Muhammad Riaz, and Muhammad Aslam. "Linear Diophantine fuzzy graphs with new decision-making approach." AIMS Mathematics 7, no. 8 (2022): 14532–56. http://dx.doi.org/10.3934/math.2022801.

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<abstract><p>The concept of linear Diophantine fuzzy set (LDFS) is a new mathematical tool for optimization, soft computing, and decision analysis. The aim of this article is to extend the notion of graph theory towards LDFSs. We initiate the idea of linear Diophantine fuzzy graph (LDF-graph) as a generalization of certain theoretical concepts including, q-rung orthopair fuzzy graph, Pythagorean fuzzy graph, and intuitionistic fuzzy graph. We extend certain properties of crisp graph theory towards LDF-graph including, composition, join, and union of LDF-graphs. We elucidate these operations with various illustrations. We analyze some interesting results that the composition of two LDF-graphs is a LDF-graph, cartesian product of two LDF-graphs is a LDF-graph, and the join of two LDF-graphs is a LDF-graph. We describe the idea of homomorphisms for LDF-graphs. We observe the equivalence relation via an isomorphism between LDF-graphs. Some significant results related to complement of LDF-graph are also investigated. Lastly, an algorithm based on LDFSs and LDF-relations is proposed for decision-making problems. A numerical example of medical diagnosis application is presented based on proposed approach.</p></abstract>

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Antonov, A. I., and V. A. Bondarenko. "Polyhedral Graphs of GRAPH PARTITIONING and COMPLETE BIPARTITE SUBGRAPH Problems." Modeling and Analysis of Information Systems 19, no. 6 (March 12, 2015): 101–6. http://dx.doi.org/10.18255/1818-1015-2012-6-101-106.

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We provide an effective description of graphs of polyhedra for GRAPH PARTITIONING and COMPLETE BIPARTITE SUBGRAPH problems. We establish the fact, that the clique number for each of this problems increases exponentially with the dimension of the space.

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Dourado, Mitre C., Priscila Petito, Rafael B. Teixeira, and Celina M. H. de Figueiredo. "Helly property, clique graphs, complementary graph classes, and sandwich problems." Journal of the Brazilian Computer Society 14, no. 2 (2008): 45–52. http://dx.doi.org/10.1590/s0104-65002008000200004.

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Zhao, Hua-An, and Wataru Mayeda. "Some problems between a w-graph and its derived graphs." Journal of the Franklin Institute 330, no. 2 (March 1993): 369–81. http://dx.doi.org/10.1016/0016-0032(93)90010-r.

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