Graph Operations Research Articles

Algorithms for 2-club cluster deletion problems using automated generation of branching rules

In the 2-Club Cluster Vertex Deletion (resp., 2-Club Cluster Edge Deletion) problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices (resp., edges) whose removal from G results in a graph in which the diameter of every connected component is at most 2. In this paper we give algorithms for 2-Club Cluster Vertex Deletion and 2-Club Cluster Edge Deletion whose running times are O⁎(3.104k) and O⁎(2.562k), respectively. Our algorithms were obtained using automated generation of branching rules. Our results improve the previous O⁎(3.303k)-time algorithm for 2-Club Cluster Vertex Deletion [Liu et al., FAW-AAIM 2012] and the O⁎(2.695k)-time algorithm for 2-Club Cluster Edge Deletion [Abu-Khzam et al., TCS 2023].

Spectral Turán problems for intersecting even cycles

Let C2k1,2k2,…,2kt denote the graph obtained by intersecting t distinct even cycles C2k1,C2k2,…,C2kt at a unique vertex. In this paper, we determine the unique graph with maximum adjacency spectral radius among all graphs on n vertices that do not contain any C2k1,2k2,…,2kt as a subgraph, for n sufficiently large. When one of the constituent even cycles is a C4, our results improve upper bounds on the Turán numbers for intersecting even cycles that follow from more general results of Füredi [21] and Alon, Krivelevich and Sudakov [1]. Our results may be seen as extensions of previous results for spectral Turán problems on forbidden even cycles C2k,k≥2 (see [8,36,46,47]).

On the Use of Geometric Deep Learning for the Iterative Classification and Down-Selection of Analog Electric Circuits

Abstract Many complex engineering systems can be represented in a topological form, such as graphs. This paper utilizes a machine learning technique called Geometric Deep Learning (GDL) to aid designers with challenging, graph-centric design problems. The strategy presented here is to take the graph data and apply GDL to seek the best realizable performing solution effectively and efficiently with lower computational costs. This case study used here is the synthesis of analog electrical circuits that attempt to match a specific frequency response within a particular frequency range. Previous studies utilized an enumeration technique to generate 43,249 unique undirected graphs presenting valid potential circuits. Unfortunately, determining the sizing and performance of many circuits can be too expensive. To reduce computational costs with a quantified trade-off in accuracy, the fraction of the circuit graphs and their performance are used as input data to a classification-focused GDL model. Then, the GDL model can be used to predict the remainder cheaply, thus, aiding decision-makers in the search for the best graph solutions. The results discussed in this paper show that additional graph-based features are useful, favorable total set classification accuracy of 80% in using only 10% of the graphs, and iteratively built GDL models can further subdivide the graphs into targeted groups with medians significantly closer to the best and containing 88.2 of the top 100 best-performing graphs on average using 25% of the graphs.

Hypergraph-based locality-enhancing methods for graph operations in Big Data applications

The need for speeding up data analytics increases inevitably due to the need for extracting valuable information from social media, data generated by smart devices with sensors, patterns of people’s communications over the web, items viewed and bought by global-scale customers, cloud applications, etc., all of which take part in the “Big Data.” Such kind of interaction data is very well represented as sparse graphs to enable the graph analytics, which requires efficient underlying kernels. The breadth-first search (BFS)-based traversal is a commonly used kernel in graph algorithms such as the betweenness centrality algorithm for centrality analysis. In this work, we focus on parallel BFS operations and propose hypergraph-based combinatorial models that aim at reducing cache misses and hence exploiting data locality during the parallel BFS operations. Our models are based on finding new vertex visit orders so that locality in accessing the data associated with vertices is exploited. Experiments on graphs arising in a wide range of applications show that our proposed models achieve on average 9% performance improvement in the CPU-based Ligra data analytics framework.

Mapping Connectivity Patterns: Degree-Based Topological Indices of Corona Product Graphs

Graph theory (GT) is a mathematical field that involves the study of graphs or diagrams that contain points and lines to represent the representation of mathematical truth in a diagrammatic format. From simple graphs, complex network architectures can be built using graph operations. Topological indices (TI) are graph invariants that correlate the physicochemical and interesting properties of different graphs. TI deal with many properties of molecular structure as well. It is important to compute the TI of complex structures. The corona product (CP) of two graphs G and H gives us a new graph obtained by taking one copy of G and V G copies of H and joining the i th vertex of G to every vertex in the i th copy of H . In this paper, based on various CP graphs composed of paths, cycles, and complete graphs, the geometric index (GA) and atom bond connectivity (ABC) index are investigated. Particularly, we discussed the corona products P s ⨀ P t , C t ⨀ C s , K t ⊙ K s , K t ⊙ P s , and P s ⊙ K t and GA and ABC index. Moreover, a few molecular graphs and physicochemical features may be predicted by considering relevant mathematical findings supported by proofs.

Investigation of dissimilar metal (stainless steel and copper) joining and the effect of offsetting and other welding parameters

Dissimilar metal joining without additional preparation like preheating and grooving using conventional arc welding processes. The challenges related to dissimilar joints are overcome by using proper heat distribution according to the heat requirement of different plates. In the present study, GTAW is used to join SS and Cu, mode of joining is welding-brazing mode on the SS side. The heat distribution is done by offsetting the arc towards the Cu side because of the high thermal diffusivity of Cu compared to stainless steel (SS). For the complete joining, the 2nd pass is done without offset. A “beach” of unmixed Cu metal along the Cu fusion line and “peninsula” of an unmixed SS in the weld metal is observed in the microstructure. The grain size variation is observed in the different zones of the weldment due to the segregation of SS particles in weldment. The joint integrity is analyzed with an optical microscope image of the bead, and the detailed analysis of weldment is done using FESEM images in BSD mode. From the microstructure porosity and Cu, penetration is observed in the SS side mushy zone. The microhardness of the weld bead is utilized to see the effect of grain size and amount of SS particles on the mechanical property of weldment in the different zone of the weld bead. The tensile test is conducted to study the joint property of the test coupon. The joint strength of Joint-7 is 225 MPa and percentage elongation is 17 percent, efficiency obtained is around 85 per cent of the Cu plate.

Hop Italian Domination in Graphs

Given a simple graph $G=(V(G),E(G))$, a function $f:V(G)\to \{0,1,2\}$ is a hop Italian dominating function if for every vertex $v$ with $f(v)=0$ there exists a vertex $u$ with $f(u)=2$ for which $u$ and $v$ are of distance $2$ from each other or there exist two vertices $w$ and $z$ for which $f(w)=1=f(z)$ and each of $w$ and $z$ is of distance $2$ from $v$. The minimum weight $\sum_{v\in V(G)}f(v)$ of a hop Italian dominating function is the hop Italian domination number of $G$, and is denoted by $\gamma_{hI}(G)$. In this paper, we initiate the study of the hop Italian domination. In particular, we establish some properties of the the hop Italian dominating function and explore the relationships of the hop Italian domination number with the hop Roman domination number \cite{Rad2,Natarajan} and with the $2$-hop domination number \cite{Canoy}. We study the concept under some binary graph operations. We establish tight bounds and determine exact values for their respective hop Italian domination numbers.

Minimum length word-representants of word-representable graphs

A graph G=(V,E) is said to be word-representable if a word w can be formed using the letters of the alphabet V such that for every pair of vertices x and y, xy∈E if and only if x and y alternate in w. Gaetz and Ji have recently introduced the notion of minimum length word-representants for word-representable graphs. They have also determined the minimum possible length of the word-representants for certain classes of graphs, such as trees and cycles.In this paper, we solved two of the three cases and partially solved the remaining case of an open problem presented in Gaetz and Ji (2020) by Gaetz and Ji. This problem asks to classify all 2-word-representable graphs based on the length of their minimum length word-representants. Furthermore, we found a relation between the maximum and minimum number of occurrences of letters in minimum length word-representants and the diameter of the graph. Finally, we gave an upper bound on the minimum possible length of a word-representant of a graph under certain word-representability preserving graph operations such as connecting two graphs by an edge and gluing two graphs in a vertex.

Graph Algebras and Derived Graph Operations

We revise our former definition of graph operations and correspondingly adapt the construction of graph term algebras. As a first contribution to a prospective research field, Universal Graph Algebra, we generalize some basic concepts and results from algebras to graph algebras. To tackle this generalization task, we revise and reformulate traditional set-theoretic definitions, constructions and proofs in Universal Algebra by means of more category-theoretic concepts and constructions. In particular, we generalize the concept of generated subalgebra and prove that all monomorphic homomorphisms between graph algebras are regular. Derived graph operations are the other main topic. After an in-depth analysis of terms as representations of derived operations in traditional algebras, we identify three basic mechanisms to construct new graph operations out of given ones: parallel composition, instantiation, and sequential composition. As a counterpart of terms, we introduce graph operation expressions with a structure as close as possible to the structure of terms. We show that the three mechanisms allow us to construct, for any graph operation expression, a corresponding derived graph operation in any graph algebra.

Effects of Dry Density and Moisture Content on the Kaolin–Brass Interfacial Shear Adhesion

Kaolin clay, with its consistent properties, fine particle size, high surface area, and extensive historical use, stands out as a reliable choice for laboratory research. This study aims to assess the interface shear adhesion behaviour between compacted clay and a metallic surface. For this purpose, a new testing approach was developed. The proposed method is simple, requires neither advanced equipment nor specialised test procedures, and, thus, represents an improvement over existing practices in this field. The experimental program focuses on determining the interface shear adhesion strength between reconstituted kaolin clay and a metallic surface. The kaolin clay testing specimens were dynamically compacted at various energy levels and moisture contents. The results indicate that the optimum moisture content is 30%, which provides the highest density to the sample and divides the compaction curve into dry and wet sides. Furthermore, the results demonstrate that the interface shear adhesion strength increases with the clay’s dry density. Conversely, there is a significant decrease in strength as the moisture content specifically rises on the wet side of the compaction curve. The adhesion behaviour was also attributed to matric suction, where high suction enhanced interfacial adhesion, while low suction weakened bonding and diminished adhesion. Additionally, this study presents a unique three-dimensional contour graph illustrating the combined effects of dry density and moisture content on the interfacial adhesion.

A fuzzy planar subgraph formation model for partitioning very large-scale integration networks

This study focuses on forming planar subgraphs in fuzzy graphs where the planarity value is not equal to 1. The paper commences by introducing the concepts of vertex-deletion (VD) and edge-deletion (ED) operations applied to fuzzy graphs. These operations aim to derive the fuzzy planar subgraphs and extensively examine the consequential significant findings. Additionally, maximal and maximum planar subgraphs in vertex-deletion (VD) and edge-deletion (ED) are presented alongside illustrative examples. A novel approach called the Planar Partition subgraph (PP subgraph) is introduced for identifying the fuzzy planar subgraphs from a fuzzy graph with an algorithm rooted in PP subgraphs for enhanced efficiency. Furthermore, the concept of thickness value in fuzzy graphs is constructed using PP-subgraphs is introduced. The discussion involves the thickness value of a fuzzy planar graph and its connection to the planarity value, supported by illustrative examples. Finally, an application made on PP subgraphs where the network needs to be partitioned into a finite number of subgraphs is demonstrated in a very large-scale integration network.

A Spatial Filter Temporal Graph Convolutional Network for decoding motor imagery EEG signals

Due to the influence of volume conduction, information redundancy exists between electrodes in electroencephalogram (EEG) signals, resulting in low spatial resolution. Furthermore, the current method of extracting EEG spatial features in Euclidean space cannot accurately depict the complex relationships among multiple electrodes, leading to poor decoding performance of EEG signals. A novel decoding framework for motor imagery EEG signals, known as the Spatial Filter Temporal Graph Convolutional Network (SF-TGCN), is proposed in this study. The spatial resolution of motor imagery EEG signals is enhanced by constructing the spatial filtering module using the Laplacian graph operator. Deep temporal features of the EEG signals are extracted using multiple layers of temporal convolutional modules, and the graph convolutional network module is employed to learn the topology of electrode connections, facilitating the decoding of motor imagery EEG signals. Superior performance is demonstrated by the proposed decoding model, achieving an average accuracy of 80.82% on the BCI IV 2a public dataset and 87.98% on a private dataset, as compared to other state-of-the-art decoding approaches.

On t-intuitionistic fuzzy graphs: a comprehensive analysis and application in poverty reduction

This paper explains the idea of t-intuitionistic fuzzy graphs as a powerful way to analyze and display relationships that are difficult to understand. The article also illustrates the ability of t-intuitionistic fuzzy graphs to establish complex relationships with multiple factors or dimensions of a physical situation under consideration. Moreover, the fundamental set operations of t-intuitionistic fuzzy graphs are proposed. The notions of homomorphism and isomorphism of t-intuitionistic fuzzy graphs are also introduced. Furthermore, the paper highlights a practical application of the proposed technique in the context of poverty reduction within a specific society. By employing t-intuitionistic fuzzy graphs, the research demonstrates the potential to address the multifaceted nature of poverty, considering various contributing factors and their interdependencies. This application showcases the versatility and effectiveness of t-intuitionistic fuzzy graphs as a tool for decision-making and policy planning in complex societal issues.

A simple [formula omitted]-approximation algorithm for Split Vertex Deletion

A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph G and weight function w:V(G)→Q≥0, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices X such that G−X is a split graph. It is easy to show that a graph is a split graph if and only if it does not contain a 4-cycle, 5-cycle, or a two edge matching as an induced subgraph. Therefore, SVD admits an easy 5-approximation algorithm. On the other hand, for every δ>0, SVD does not admit a (2−δ)-approximation algorithm, unless P=NP or the Unique Games Conjecture fails.For every ϵ>0, Lokshtanov, Misra, Panolan, Philip, and Saurabh (Lokshtanov et al., 2020) recently gave a randomized(2+ϵ)-approximation algorithm for SVD. In this work we give an extremely simple deterministic (2+ϵ)-approximation algorithm for SVD.

Potential effect of managing connectivity to contain disease spread among free‐ranging wild boar (Sus scrofa) in disparate landscapes

Abstract Landscape connectivity is a major factor shaping the spread of pathogens in wildlife populations. By managing connectivity, transmission pathways can be broken and disease spread be contained, particularly in the early phases of an outbreak. Having witnessed recent outbreaks of African Swine Fever in free‐ranging wild boar (Sus scrofa) in Belgium, Germany and Italy, offices for disease control are on the alert also in other western European countries. This study investigates the potential effect of managing landscape connectivity to contain disease spread among free‐ranging wild boar in disparate landscapes. It involves research into (1) the ease with which wildlife corridors can be blocked for wild boar, (2) the connectivity of wild boar habitat and (3) the impact of landscape fragmentation on connectivity management. This is addressed by carrying out GIS analyses and performing graph operations on the wild boar networks in different biogeographical regions of Switzerland. The results of doing research into the three above‐mentioned objects show that, regarding the first, most wildlife corridors are hard to block for wild boar, because their features or location make fencing difficult. Regarding the second, the wild boar habitat is connected. Opening wildlife passages that are currently under construction may allow wild boar to disperse to hitherto uncolonized areas. Regarding the third, all wild boar networks could be partially decomposed by blocking the easy to block corridors and closing the passages. Network decomposition would be easiest to achieve in the region where the built infrastructure is most abundant. All over Switzerland, the potential epidemic size could be reduced by 25% when blocking the minimal set of corridors and passages that cut the networks to non‐decomposable components. This study suggests that (a) combining connectivity analysis with fragmentation analysis is key to explaining why a specific measure of disease containment is more effective in one landscape than in the other, (b) complementing the permeability model with a species distribution model is essential to identify connected habitat patches for the species of concern and (c) connectivity metrics should consider also the surface area of occupied habitat patches and relative abundance of the species of concern.

Albertson (Alb) spectral radii and Albertson (Alb) energies of graph operation.

The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest absolute eigenvalue of the adjacency matrix is represented by the spectral radius of the graph. Both molecular computing and computer science have uses for graph energies and spectral radii. The Albertson (Alb) energies and spectral radii of generalized splitting and shadow graphs constructed on any regular graph is the main focus of this study. The only thing that may be disputed is the comparison of the (Alb) energies and (Alb) spectral radii of the newly formed graphs to those of the base graph. By concentrating on splitting and shadow graph, we compute new correlations between the Alb energies and spectral radius of the new graph and the prior graph.

Thermal performance of energy barrettes in an urban environment

Thermal performance of energy barrettes in an urban environment

A Comparative Study of Fuzzy Domination and Fuzzy Coloring in an Optimal Approach

An optimal network refers to a computer or communication network designed, configured, and managed to maximize efficiency, performance, and effectiveness while minimizing cost and resource utilization. In a network design and management context, optimal typically implies achieving the best possible outcomes between various factors. This research investigated the use of fuzzy graph edge coloring for various fuzzy graph operations, and it focused on the efficacy and efficiency of the fuzzy network product using the minimal spanning tree and the chromatic index of the fuzzy network product. As a network made of nodes and vertices, measurement with vertices is a parameter for domination, and edge measurement is a parameter for edge coloring, so we used these two parameters in the algorithm. This paper aims to identify an optimal network that can be established using product outcomes. This study shows a way to find an optimal fuzzy network based on comparative optimal parameter domination and edge coloring, which can be elaborated with applications. An algorithm was generated using an optimal approach, which was subsequently implemented in the form of applications.

Streaming deletion problems parameterized by vertex cover

Streaming is a model where an input graph is provided one edge at a time, instead of being able to inspect it at will. In this work, we take a parameterized approach by assuming a vertex cover of the graph is given, building on work of Bishnu et al. [COCOON 2020]. We show the further potency of combining this parameter with the Adjacency List streaming model to obtain results for vertex deletion problems. This includes kernels, parameterized algorithms, and lower bounds for the problems of Π-free Deletion, H-free Deletion, and the more specific forms of Cluster Vertex Deletion and Odd Cycle Transversal. We focus on the complexity in terms of the number of passes over the input stream, and the memory used. This leads to a pass/memory trade-off, where a different algorithm might be favourable depending on the context and instance. We also discuss implications for parameterized complexity in the non-streaming setting.

Scheduling operations in a large hospital by multiple agents

The scheduling of operations in a large hospital is performed jointly by several groups of people, each with its own objective and constraints. It is a two-phase process, starting with the allocation of operating rooms to wards, and followed by the scheduling of operations in each operating room of the hospital on each day. The final schedule must satisfy all inter-ward hard constraints, such as the allocation of anesthetists, nurses, and equipment to operations that are taking place in parallel, and ideally, it should also address soft constraints such as taking the urgency and complexity of operations into consideration.This study contributes to the ongoing effort of adapting multi-agent optimization models and algorithms to real-world applications by modeling the problems in both phases as distributed constraint optimization problems (DCOPs), with different properties. The first phase includes partially cooperative ward-representing agents, allocating operating rooms for daily usage among themselves. In the second phase, ward-representing agents interact with agents representing constraining elements, in order to generate daily operation schedules for each operating room, thus forming a unique bipartite constraint graph. On one side are the ward representatives, while on the other are the agents representing the constraining resources. Each agent has a non-trivial local problem to solve, and its solution serves as the proposed assignment in the distributed algorithm.The study begins by discussing the properties required of the algorithms needed to solve the two phases. It then proposes adjustments to existing distributed partially cooperative algorithms and local search algorithms to solve these problems, and compares the results of different variants of these algorithms. The results obtained for both phases emphasize that successful collaboration is predicated on two requirements: that agents hold consistent information regarding their peers’ states and that the degree of exploration undertaken by the algorithm is restricted in order to produce high-quality solutions.