Graph Operations Research Articles

DGCNN approach links metagenome-derived taxon and functional information providing insight into global soil organic carbon

Metagenomics can provide insight into the microbial taxa present in a sample and, through gene identification, the functional potential of the community. However, taxonomic and functional information are typically considered separately in downstream analyses. We develop interpretable machine learning (ML) approaches for modelling metagenomic data, combining the biological representation of species with their associated genetically encoded functions within models. We apply our methods to investigate soil organic carbon (SOC) stocks. First, we combine a diverse global set of soil microbiome samples with environmental data, improving the predictive performance of classic ML and providing new insights into the role of soil microbiomes in global carbon cycling. Our network analysis of predictive taxa identified by classical ML models provides context for their ecological significance, extending the focus beyond just the most predictive taxa to ‘hidden’ features within the model that might be considered less predictive using standard methods for explainability. We next develop unique graph representations for individual microbiomes, linking microbial taxa to their associated functions directly, enabling predictions of SOC via deep graph convolutional neural networks (DGCNNs). Interpretation of the DGCNNs distinguished between the importance of functions of key individual species, providing genome sequence differences, e.g., gene loss/acquisition, that associate with SOC. These approaches identify several members of the Verrucomicrobiaceae family and a range of genetically encoded functions, e.g., related to carbohydrate metabolism, as important for SOC stocks and effective global SOC predictors. These relatively understudied but widespread organisms could play an important role in SOC dynamics globally.

Periodic diffraction from an aperiodic monohedral tiling - the Spectre tiling. Addendum.

This article describes the diffraction pattern (2-periodic Fourier transform) from the vertices of a large patch of the recently discovered `Spectre' tiling - a strictly chiral aperiodic monotile. It was reported recently that the diffraction pattern of the related weakly chiral aperiodic `Hat' monotile was 2-periodic with chiral plane-group symmetry p6 [Kaplan et al. (2024). Acta Cryst. A80, 72-78]. The diffraction periodicity arises because the Hat tiling is a systematic aperiodic deletion of vertices from the 2-periodic hexagonal mta tiling. Despite the similarity of the Hat and Spectre tilings, the Spectre tiling is not aligned with a 2-periodic lattice, and its diffraction pattern is non-periodic with chiral point symmetry 6 about the origin.

Outdegree equitable domination number of certain graph operators

For a vertex u in the dominating set D of a graph G, the number of edges from u to V-D is called the outdegree of u with respect to D, d°D, G(u). A dominating set D is called the outdegree equitable dominating set if the absolute value of the differences of outdegrees of any two vertices in D is at most one. The minimum cardinality of an outdegree equitable dominating set of G is called the outdegree equitable domination number of G, γoe(G). In this paper, we study the outdegree equitable domination number of certain graph operators such as complement, double graph, mycielskian and subdivision of graph.

Scalable Metadata-Hiding for Privacy-Preserving IoT Systems

Modern cloud-based IoT services comprise an integrator service and several device vendor services. The vendor services enable users to remotely control their devices, while the integrator serves as a central intermediary, offering a unified interface for managing devices from different vendors. Although such a model is quite beneficial for IoT services to evolve quickly, it also creates a serious privacy concern: the vendor and integrator services observe all interactions between users and devices. Toward this, we propose Mohito, a privacy-preserving IoT system that hides such interactions from both the integrator and the vendors. In Mohito, we protect both the interaction data and the metadata, so that no one learns which user is communicating with which device. By utilizing oblivious key-value storage as a primitive and leveraging the unique communication graph of IoT services, we build a scalable protocol specialized in handling large concurrent traffic, a common demand in IoT systems. Our evaluation shows that Mohito can achieve up to 600x more throughput than the state-of-the-art general-purpose systems that provide similar security guarantees.

Fractional Vertex Cover Reliability of Graphs

Let G be a graph and let 0 ≤ p , q and p + q ≤ 1 . Suppose that each vertex of G gets a weight of 1 with probability p , 1 2 with probability q , and 0 with probability 1 − p − q , and vertex weight probabilities are independent. The \textit{fractional vertex cover reliability} of G , denoted by FRel ( G ; p , q ) , is the probability that the sum of weights at the end-vertices of every edge in G is at least 1 . In this article, we first provide various computational formulas for FRel ( G ; p , q ) considering general graphs, basic graphs, and graph operations. Secondly, we determine the graphs which maximize FRel ( G ; p , q ) for all values of p and q in the classes of trees, connected unicyclic and bicyclic graphs with fixed order, and determine the graphs which minimize it in the classes of trees and connected unicyclic graphs with fixed order. Our results on optimal graphs extend some known results in the literature about independent sets, and the tools we developed in this paper have the potential to solve the optimality problem in other classes of graphs as well.

A Study on Zagreb Indices of Vertex-Switching for Special Graph Classes

Many graph theorists have studied graph operations due to their applications and the advantages with heavy calculations. In a recent paper, the vertex-switching operation is analyzed, and some vertex-switched graphs are determined for some graph classes. This paper calculates the first Zagreb index and the second Zagreb index of vertex-switched star, complete bipartite, and tadpole graphs. It finally discusses the need for further research.

On the Independent Uphill Domination Polynomial of a Graph

Objectives: The concept of independent uphill domination polynomial is newly defined to deal with the present graph theory in this paper. It is characterized and analyzed within various contexts and types of graphs, its roots are discovered, its relation to different graph transforms is discussed and the study of this graph can help reveal new structures of graphs. Methods: The independent uphill domination polynomial is defined as is an independent uphill dominating set of size in . The uphill paths are used in identifying the criterion of independent uphill dominating sets. The computation of specific values of the polynomial is performed for basic graphs, and the polynomial’s response to various flip operations is observed. Findings: Checking the standard graphs for some properties of the independent uphill dominance polynomial shows the features of the independent uphill dominance polynomial as an uphill dominance polynomial of a certain type of digraph. Families of polynomials can be best understood when it comes to stability and characteristics by analyzing their roots on different graphical planes. They demonstrate how the structure of the graph and the polynomial change as a result of the different graph operations. Novelty: Incorporating uphill paths into independent dominating sets extends common domination polynomials and provides a distinct perspective distinct from previous investigations on graph dominance. Hence, this work enriches the subject and suggests new avenues for studying related polynomials and their uses by encompassing more structural properties of graphs. Keywords: Domination, Polynomial, Uphill domination, Independent domination, Root, Book graph, Independent uphill domination polynomial

Structural Parameterizations for Two Bounded Degree Problems Revisited

We revisit two well-studied problems, Bounded Degree Vertex Deletion and Defective Coloring , where the input is a graph G and a target degree Δ, and we are asked either to edit or partition the graph so that the maximum degree becomes bounded by Δ. Both problems are known to be parameterized intractable for the most well-known structural parameters, such as treewidth. We revisit the parameterization by treewidth, as well as several related parameters and present a more fine-grained picture of the complexity of both problems. In particular, we present the following: — Both problems admit straightforward DP algorithms with table sizes \((\Delta +2)^\mathrm{tw}\) and \((\chi _\mathrm{d}(\Delta +1))^{\mathrm{tw}}\) , respectively, where tw is the input graph’s treewidth and \(\chi _\mathrm{d}\) the number of available colors. We show that, under the SETH, both algorithms are essentially optimal, for any non-trivial fixed values of \(\Delta , \chi _\mathrm{d}\) , even if we replace treewidth by pathwidth. Along the way, we obtain an algorithm for Defective Coloring with complexity quasi-linear in the table size, thus settling the complexity of both problems for treewidth and pathwidth. — Given that the standard DP algorithm is optimal for treewidth and pathwidth, we then go on to consider the more restricted parameter tree-depth. Here, previously known lower bounds imply that, under the ETH, Bounded Vertex Degree Deletion and Defective Coloring cannot be solved in time \(n^{o(\sqrt [4]{\mathrm{td}})}\) and \(n^{o(\sqrt {\mathrm{td}})}\) , respectively, leaving some hope that a qualitatively faster algorithm than the one for treewidth may be possible. We close this gap by showing that neither problem can be solved in time \(n^{o(\mathrm{td})}\) , under the ETH, by employing a recursive low tree-depth construction that may be of independent interest. — Finally, we consider a structural parameter that is known to be restrictive enough to render both problems FPT: vertex cover. For both problems the best known algorithm in this setting has a super-exponential dependence of the form \(\mathrm{vc}^{\mathcal {O}(\mathrm{vc})}\) . We show that this is optimal, as an algorithm with dependence of the form \(\mathrm{vc}^{o(\mathrm{vc})}\) would violate the ETH. Our proof relies on a new application of the technique of d -detecting families introduced by Bonamy et al. [ToCT 2019]. Our results, although mostly negative in nature, paint a clear picture regarding the complexity of both problems in the landscape of parameterized complexity, since in all cases we provide essentially matching upper and lower bounds.

Bloch varieties and quantum ergodicity for periodic graph operators

Bloch varieties and quantum ergodicity for periodic graph operators

Strong incidence domination in some operations of fuzzy incidence graphs and application in security allocation

Strong incidence domination in some operations of fuzzy incidence graphs and application in security allocation

A bivariate simultaneous pollutant forecasting approach by Unified Spectro-Spatial Graph Neural Network (USSGNN) and its application in prediction of O3 and NO2 for New Delhi, India

Declining urban air quality affects socioeconomic stability, public health, and ecosystems and is demanding attention of the administration to address environmental sustainability goals. Given the effects of ozone, a greenhouse gas, on local climate and health, this study introduces a Unified Spectro-Spatial Graph Neural Network (USS-GNN) designed for simultaneous 24-hour forecasting of ozone and its precursor, nitrogen-dioxide, while addressing their chemical interactions and spatiotemporal dynamics. This model exploits the graph structure of atmospheric dynamics and mines high-level spatial, spectral, and physical features from atmospheric data through a Dot Product Edge Attention mechanism and a location-aware graph feature rewiring technique. The proposed model is developed for Indian capital city New Delhi, utilizes hourly observations for the years 2021 and 2022 and achieved R2 values of 0.650 and 0.618, RMSE of 13.950 and 16.120 μg/m3, MAE of 10.730 and 12.930 μg/m3 for ozone and nitrogen-dioxide respectively, outperforming state-of-the-art models. The model’s forecast analysis identified error-prone areas, effects of local meteorology, and pollutant interdependencies. An ablation study further detailed the impacts of graph operations on forecasts. Moreover, this study promotes the utility of bivariate modeling frameworks in improving urban pollution monitoring and supporting sustainable city management through data-driven policy implementations.

Pre-training enhanced spatio-temporal graph neural network for predicting influent water quality and flow rate of wastewater treatment plant: Improvement of forecast accuracy and analysis of related factors

Efficient management of wastewater treatment plants (WWTPs) necessitates accurate forecasting of influent water quality parameters (WQPs) and flow rate (Q) to reduce energy consumption and mitigate carbon emissions. The time series of WQPs and Q are highly non-linear and influenced by various factors such as temperature (T) and precipitation (Precip). Conventional models often struggle to account for long-term temporal patterns and overlook the complex interactions of parameters within the data, leading to inaccuracies in detecting WQPs and Q. This work introduced the Pre-training enhanced Spatio-Temporal Graph Neural Network (PT-STGNN), a novel methodology for accurately forecasting of influent COD, ammonia nitrogen (NH3-N), total phosphorus (TP), total nitrogen (TN), pH and Q in WWTPs. PT-STGNN utilizes influent data of the WWTP, air quality data and meteorological data from the service area as inputs to enhance prediction accuracy. The model employs unsupervised Transformer blocks for pre-training, with efficient masking strategies to effectively capture long-term historical patterns and contextual information, thereby significantly boosting forecasting accuracy. Furthermore, PT-STGNN integrates a unique graph structure learning mechanism to identify dependencies between parameters, further improving the model's forecasting accuracy and interpretability. Compared with the state-of-the-art models, PT-STGNN demonstrated superior predictive performance, particularly for a longer-term prediction (i.e., 12 h), with MAE, RMSE and MAPE at 12-h prediction horizon of 2.737 ± 0.040, 4.209 ± 0.060 and 13.648 ± 0.151 %, respectively, for the algebraic mean of each parameter. From the results of graph structure learning, it is observed that there are strong dependencies between NH3-N and TN, TP and Q, as well as Precip, etc. This study innovatively applies STGNN, not only offering a novel approach for predicting influent WQPs and Q in WWTPs, but also advances our understanding of the interrelationships among various parameters, significantly enhancing the model's interpretability.

Modification problems toward proper (Helly) circular-arc graphs

We present a 9k⋅nO(1)-time algorithm for the proper circular-arc vertex deletion problem, resolving an open problem of van 't Hof and Villanger [Algorithmica 2013] and Crespelle et al. [Computer Science Review 2023]. Our structural study also implies parameterized algorithms for modification problems toward proper Helly circular-arc graphs.

An intuitionistic fuzzy graph’s variation coefficient measure with application to selecting a reliable alliance partner

Group decision-making (GDM) is crucial in various components of graph theory, management science, and operations research. In particular, in an intuitionistic fuzzy group decision-making problem, the experts communicate their preferences using intuitionistic fuzzy preference relations (IFPRs). This approach is a way that decision-makers rank or select the most desirable alternatives by gathering criteria-based information to estimate the best alternatives using a wider range of knowledge and experience. This article proposes a new statistical measure in a fuzzy environment when the data is ambiguous or unreliable to solve a decision-making problem. This study uses the variation coefficient measure combined with intuitionistic fuzzy graphs (IFG) and Laplacian energy (LE) to solve a GDM problem that utilizes intuitionistic fuzzy preference relations (IFPRs) to select a reliable alliance partner. Initially, the Laplacian energy determines the weight of individual standards, and the obtained weight average further estimates the overall criterion weight vector. We establish the authority criteria weights using the variation coefficient measure and then ultimately rank the alternatives for each criterion using the same measure. We examine four distinct companies Alpha, Beta, Delta, and Zeta to conduct a realistic GDM to choose which alliance partner would be ideal. We successfully implemented the suggested technique, determining that Alpha satisfies company standards and is ranked first among other companies. Moreover, this technique is useful for all kinds of Intuitionistic fuzzy group decision-making problems to select optimal ones.

Polynomial Representation and Degree Sequence of Graphs Resulting from Some Graph Operations

Let $G = (V(G),E(G))$ be a graph with degree sequence $\langle d_1,d_2, \cdots, d_n \rangle$, where $d_1 \ge d_2 \ge \cdots \ge d_n$. The polynomial representation of $G$ is given by $f_G(x) = \displaystyle \sum_{i=1}^n x^{d_i} = \sum_{k=1}^{\Delta(G)}a_kx^{k}$, where $a_k$ is the number of vertices of $G$ having degree $k$ for each $i = 1,2,\cdots n = \Delta(G)$. In this paper, we give the polynomial representation of the complement and line graph of a graph, the shadow graph, complementary prism, edge corona, strong product, symmetric product, and disjunction of two graphs.

Anomaly detection with dual-channel heterogeneous graph based on hypersphere learning

Graph anomaly detection is essential for identifying irregular patterns and outliers within complex network structures in domains like social networks, cybersecurity, finance, and transportation systems. It helps detect security breaches, fraud, and errors, improving decision-making and system reliability. Although current methodologies have advanced the unsupervised detection of graph anomalies, they frequently fail to fully address the nuanced specificities of graph anomalies, such as anomalous nodes, edges, and subgraphs. To overcome this limitation, the study presents MulDualGNN, a unique dual-channel heterogeneous graph anomaly detection framework. MulDualGNN incorporates a global substructure-aware GNN and a local substructure-aware GNN to capture both global and local substructure properties for accurate anomaly detection. Our model incorporates a multi-hypersphere learning target function, which includes macroscopic and mesoscopic hyperspheres. These hyperspheres measure abnormal nodes that deviate from most normal nodes in the entire graph and community structure, respectively. To overcome the model collapse problem in multi-hypersphere learning, our model utilizes the EmbSim similarity function to optimize the training target. The effectiveness and performance advantages of the proposed method are evaluated through extensive experiments on five datasets. The results demonstrate the superior performance of our approach in graph anomaly detection tasks.

Extremal graph of super line graph operation via generalized Randić index

Graph operations enable the modification and transformation of graphs, enhancing data representation as well as effectiveness in fields such as computer science, mathematics, and data analysis. Super line graph operations are a concept of advanced graph theory that is used in advanced mathematics and computer science. They are used to solve specific mathematical problems, especially in graph theory and Combinatorics. Topological numbers are numerical values connected with graph structures that are used to analyze structural characteristics and solve problems in fields such as chemistry and mathematics. They are important for simplifying complex data and enabling quantitative analysis. The general Randić number of some super line graph operations of number two with a three-diameter is investigated in this study. The pendant vertices are inserted without disturbing their diameter. Seven graphs of order five, each with a diameter of three, can be used to create twenty-three generalized super line graphs.

Hidden Semi-Markov Models for Semantic-Graph Language Modeling

Semantic communication is expected to play a critical role in reducing traffic load in future intelligent large-scale sensor networks. With advances in Machine Learning (ML) and Deep Learning (DL) techniques, design of semantically-aware systems has become feasible in recent years. This work focuses on improving the reliability of the semantic information represented in a graph-based language that has been recently proposed. Inaccuracies in the representation of the semantic information can arise due to multiple factors, such as algorithmic shortcomings or sensory errors, decreasing the performance of the semantic extractor. This study aims to model the temporal evolution of semantic information, represented using the graph language, to enhance its reliability. Each unique graph configuration is treated as a distinct state, leading to a Hidden Semi-Markov Model (HSMM) defined over the state space of the graph configurations. The HSMM formulation enables the integration of prior knowledge on the semantic signal into the graph sequences, enhancing the accuracy in identifying semantic innovations. Within the HSMM framework, algorithms designed for graph smoothing, semantic information fusion, and model learning are introduced. The efficacy of these algorithms in improving the reliability of the extracted semantic-graphs is demonstrated through simulations and video streams generated in the CARLA simulation environment.

Versatile Graph Neural Networks Toward Intuitive Human Activity Understanding.

Benefiting from the advanced human visual system, humans naturally classify activities and predict motions in a short time. However, most existing computer vision studies consider those two tasks separately, resulting in an insufficient understanding of human actions. Moreover, the effects of view variations remain challenging for most existing skeleton-based methods, and the existing graph operators cannot fully explore multiscale relationship. In this article, a versatile graph-based model (Vers-GNN) is proposed to deal with those two tasks simultaneously. First, a skeleton representation self-regulated scheme is proposed. It is among the first trials that successfully integrate the idea of view adaptation into a graph-based human activity analysis system. Next, several novel graph operators are proposed to model the positional relationships and learn the abstract dynamics between different human joints and parts. Finally, a practical multitask learning framework and a multiobjective self-supervised learning scheme are proposed to promote both the tasks. The comparative experimental results show that Vers-GNN outperforms the recent state-of-the-art methods for both the tasks, with the to date highest recognition accuracies on the datasets of NTU RGB + D (CV: 97.2%), UWA3D (88.7%), and CMU (1000 ms: 1.13).

The local vertex anti-magic coloring for certain graph operations

This work proves the local vertex anti-magic coloring of even regular circulant bipartite graphs C(m;L). Let G be either Kr,r or Kr,r−F, F is a 1-factor. Also, we discover the local vertex anti-magic coloring for union of bipartite graphs; join graphs G∨H, where H∈{Or,Kr,Cr,Kr,s}; and the upper bound of corona product G⊙Or. It was a problem Lau and Shiu (2023) [1] that: For any G1 and G2, determine χℓva(G1×G2). We give partial answer to this problem by proving the followings:1.χℓva(C2m×C2n);2.χℓva(C2m+1×C2n+2); and3.χℓva(P3×H), where H∈{Kr,Km,m}.