Neighborhood Of Vertex Research Articles

On finding short reconfiguration sequences between independent sets

Token Sliding Optimization asks whether there exists a sequence of at most ℓ steps that transforms independent set S into T, where at each step a token slides to an unoccupied neighboring vertex (while maintaining independence). In Token Jumping Optimization, we are instead allowed to jump from a vertex to any unoccupied vertex. Both problems are known to be FPT when parameterized by ℓ on nowhere dense graphs. In this work, we show that both problems are FPT for parameter k+ℓ+d on d-degenerate graphs as well as for parameter |M|+ℓ+Δ on graphs having a modulator M to maximum degree Δ. We complement these results by showing that for parameter ℓ both problems become hard already on 2-degenerate graphs. Finally, we show that using such families one can obtain a unified algorithm for the standard Token Jumping problem parameterized by k on degenerate and nowhere dense graphs.

Explicit 3D reconstruction from images with dynamic graph learning and rendering-guided diffusion

High-quality 3D reconstruction is becoming increasingly important in a variety of fields. Recently, implicit representation methods have made significant progress in image-based 3D reconstruction. However, these methods tend to yield entangled neural representations which lack support for standard 3D pipelines, and their reconstruction results usually experience a sharp drop in quality when input views are reduced. To obtain high-quality 3D content, we propose an explicit 3D reconstruction method that directly extracts textured meshes from images and remains robust using reduced input views. Our central components include a dynamic graph convolutional network (GCN) and a rendering-guided diffusion model. The dynamic GCN aims to improve mesh reconstruction quality by effectively aggregating features from vertex neighborhoods. The aggregation is accelerated through sampling geometric-related neighbors with different SDF signs, which gradually converges in quantity during training. The rendering-guided diffusion model learns prior distributions for unseen regions to improve reconstruction performance using sparse-view inputs. It uses the rendered image under an interpolated camera pose as conditioned input and its diffusion strength can be controlled with the rendering loss of explicit reconstruction. In addition, the rendering-guided diffusion model can be jointly trained to generate plausible novel views with 3D consistency. Experiments demonstrate that our method can produce high-quality explicit reconstruction results and maintain realistic reconstruction using sparse-view inputs.

Theoretical studies of the [formula omitted]-strong Roman domination problem

The concept of Roman domination has been a subject of intrigue for more than two decades, with the fundamental Roman domination problem standing out as one of the most significant challenges in this field. This article studies a practically motivated generalization of this problem, known as the k-strong Roman domination. In this problem variant, defenders within a network are tasked with safeguarding any k vertices simultaneously, under multiple attacks. The aim is to find a feasible mapping that assigns a (integer) weight to each vertex of the input graph with a minimum sum of weights across all vertices. A function is considered feasible if any non-defended vertex, i.e. one labeled by zero, is protected by at least one of its neighboring vertices labeled by at least two. Furthermore, each defender ensures the safety of a non-defended vertex by imparting a value of one to it while retaining a one for themselves. To the best of our knowledge, this paper represents the first theoretical study on this problem. The study presents results for general graphs, establishes connections between the problem at hand and other domination problems, and provides exact values and bounds for specific graph classes, including complete graphs, paths, cycles, complete bipartite graphs, grids, and a few selected classes of convex polytopes. Additionally, an attainable lower bound for general cubic graphs is provided.

Some New Results on Domination and Independent Dominating Set of Some Graphs

One area of graph theory that has been studied in great detail is dominance in graphs. Applications for dominating sets are numerous. In wireless networking, dominant sets are used to find effective paths inside ad hoc mobile networks. They have also been used in the creation of document summaries and safe electrical grid systems. A set <I>S</I>⊆<I>V</I> is said to be dominating set of <I>G</I> if for every <i>v </i>є <I>V</I>-<I>S</I> there exists a vertex <i>u</i> є <I>S</I> such that <i>uv</i> є <I>E</I>. The dominance number of <I>G</I>, represented by <i>γ</i>(<I>G</I>), is the lowest cardinality of vertices among the dominating set of <I>G</I>. A classic NP-complete decision problem in computational complexity theory determines whether, given a graph <I>G</I> and input <I>K</I>, <i>γ</i>(<I>G</I>) ≤ <I>K</I>. This is known as the dominating set issue. Consequently, it is thought that calculating <i>γ</i>(<I>G</I>) for each given graph <I>G</I> may not be possible to do with a feasible algorithm. In addition to efficient approximation tactics, there exist efficient exact techniques for various graph classes. If there are no neighboring vertices in a subset <I>S</I>, then <I>S</I>⊆<I>V</I> is an independent set. Additionally, the empty set and the subset with just one vertex are independent. An independent dominating set of <I>G</I> is a set <I>S</I> of vertices in a graph <I>G</I> that is both an independent and a dominating set of <I>G</I>. This paper's primary goal is to investigate the dominance and independent dominating set of many graphs, including the line graph, the alternate triangular belt graph, the bistar graph, the triangular snake graph, and others.

Binary rat swarm optimizer algorithm for computing independent domination metric dimension problem

In this article, we look at the NP-hard problem of determining the minimum independent domination metric dimension of graphs. A vertex set B of a connected graph G(V,E) resolves G if every vertex of G is uniquely recognized by its vector of distances to the vertices in B. If there are no neighboring vertices in a resolving set B of G, then B is independent. Every vertex of G that does not belong to B must be a neighbor of at least one vertex in B for a resolving set to be dominant. The metric dimension of G, independent metric dimension of G, and independent dominant metric dimension of G are, respectively, the cardinality of the smallest resolving set of G, the minimal independent resolving set, and the minimal independent domination resolving set. We propose the first attempt to use a binary version of the Rat Swarm Optimizer Algorithm (BRSOA) to heuristically calculate the smallest independent dominant resolving set of graphs. The search agent of BRSOA are binary-encoded and used to identify which one of the vertices of the graph belongs to the independent domination resolving set. The feasibility is enforced by repairing search agent such that an additional vertex created from vertices of G is added to B, and this repairing process is repeated until B becomes the independent domination resolving set. Using theoretically computed graph findings and comparisons to competing methods, the proposed BRSOA is put to the test. BRSOA surpasses the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWOA), the binary Gravitational Search Algorithm (BGSA), and the binary Moth-Flame Optimization (BMFO), according to computational results and their analysis.

Learning Aligned Vertex Convolutional Networks for Graph Classification.

Graph convolutional networks (GCNs) are powerful tools for graph structure data analysis. One main drawback arising in most existing GCN models is that of the oversmoothing problem, i.e., the vertex features abstracted from the existing graph convolution operation have previously tended to be indistinguishable if the GCN model has many convolutional layers (e.g., more than two layers). To address this problem, in this article, we propose a family of aligned vertex convolutional network (AVCN) models that focus on learning multiscale features from local-level vertices for graph classification. This is done by adopting a transitive vertex alignment algorithm to transform arbitrary-sized graphs into fixed-size grid structures. Furthermore, we define a new aligned vertex convolution operation that can effectively learn multiscale vertex characteristics by gradually aggregating local-level neighboring aligned vertices residing on the original grid structures into a new packed aligned vertex. With the new vertex convolution operation to hand, we propose two architectures for the AVCN models to extract different hierarchical multiscale vertex feature representations for graph classification. We show that the proposed models can avoid iteratively propagating redundant information between specific neighboring vertices, restricting the notorious oversmoothing problem arising in most spatial-based GCN models. Experimental evaluations on benchmark datasets demonstrate the effectiveness.

Automatic labeling of 3D facial acupoint landmarks

<p>As special marks on a human face, facial landmarks reflect the facial features of various parts of the face, which is crucial in biomedicine and medical imaging. In addition, facial landmarks are also important features in computer vision such as face detection, face recognition, facial pose estimation, and facial animation. In this paper, we construct a 3D facial acupoint annotated dataset by labeling 37 facial acupoints on 846 neutral face triangle mesh on the FaceScape dataset. Based on these annotated data, we use a feature template matching method to realize the automatic annotation of 37 acupoints on triangle meshes. We used 40 meshes as the training set to extract the geometric patterns of 3D acupoints and then measured the performance of the automatic labeling algorithm on 20 meshes and 806 meshes as the test sets. In the training process, we extract the tangent plane for each landmark, project the neighbor vertices of the landmark to the tangent plane, and construct the feature image with <em>R</em> × <em>R</em> resolution through the bounding box of the projected points. In the testing process, we use the pattern images extracted during training to find the average features and use them as a guide to optimize the predicted landmarks. The experimental results show that our automatic acupoint labeling method has achieved good results.<strong></strong></p>

First passage time interdiction in Markov chains

We introduce a new variant of the network interdiction problem with a Markovian evader that randomly chooses a neighboring vertex in each step to build their path from designated source(s) to terminal(s). The interdictor's goal is to maximize the evader's minimum expected first passage time. We establish sufficient conditions for the interdiction to not be counter-productive, prove that the problem is NP-hard, and demonstrate the model's usefulness by solving a mixed-integer programming formulation on a test bed of social networks.

Levenberg–Marquardt deep neural watermarking for 3D mesh using nearest centroid salient point learning

Watermarking is one of the crucial techniques in the domain of information security, preventing the exploitation of 3D Mesh models in the era of Internet. In 3D Mesh watermark embedding, moderately perturbing the vertices is commonly required to retain them in certain pre-arranged relationship with their neighboring vertices. This paper proposes a novel watermarking authentication method, called Nearest Centroid Discrete Gaussian and Levenberg–Marquardt (NCDG–LV), for distortion detection and recovery using salient point detection. In this method, the salient points are selected using the Nearest Centroid and Discrete Gaussian Geometric (NC–DGG) salient point detection model. Map segmentation is applied to the 3D Mesh model to segment into distinct sub regions according to the selected salient points. Finally, the watermark is embedded by employing the Multi-function Barycenter into each spatially selected and segmented region. In the extraction process, the embedded 3D Mesh image is extracted from each re-segmented region by means of Levenberg–Marquardt Deep Neural Network Watermark Extraction. In the authentication stage, watermark bits are extracted by analyzing the geometry via Levenberg–Marquardt back-propagation. Based on a performance evaluation, the proposed method exhibits high imperceptibility and tolerance against attacks, such as smoothing, cropping, translation, and rotation. The experimental results further demonstrate that the proposed method is superior in terms of salient point detection time, distortion rate, true positive rate, peak signal to noise ratio, bit error rate, and root mean square error compared to the state-of-the-art methods.

A large-scale graph partition algorithm with redundant multi-order neighbor vertex storage

Recently, graph data become increasingly important and larger in many fields, and many distributed graph computing systems have been proposed to deal with large-scale graphs. The graph partition algorithm is the basis of these systems. In many graph applications, a vertex updates its feature by aggregating information of its multi-order neighboring vertices, and the existing graph partition algorithms often lead to heavy communication cost. This paper proposes a graph partition algorithm to store the multi-order neighbor vertices redundantly to avoid the communication requirement. It first formulates the problem of graph partitioning with redundant multi-order neighbor vertices storage as an optimization problem, and then proposes PARN (partition algorithm with redundant multi-order neighbor) algorithm to tackle it. The PARN algorithm consists of initial partition and vertex migration based on genetic algorithm, and it classifies the vertices of a partition into primary and auxiliary vertices. The auxiliary vertices required for the primary vertices in a partition are stored in the same partition, so the primary vertices don't need to fetch the information from the other partitions. The experiment results show that the proposed algorithm is comparable to Hash, GAP, GNP, and QCLP algorithms in terms of load balance and produces fewer auxiliary vertices than these algorithms.

Cavity approach for the approximation of spectral density of graphs with heterogeneous structures.

Graphs have become widely used to represent and study social, biological, and technological systems. Statistical methods to analyze empirical graphs were proposed based on the graph's spectral density. However, their running time is cubic in the number of vertices, precluding direct application to large instances. Thus, efficient algorithms to calculate the spectral density become necessary. For sparse graphs, the cavity method can efficiently approximate the spectral density of locally treelike undirected and directed graphs. However, it does not apply to most empirical graphs because they have heterogeneous structures. Thus, we propose methods for undirected and directed graphs with heterogeneous structures using a new vertex's neighborhood definition and the cavity approach. Our methods' time and space complexities are O(|E|h_{max}^{3}t) and O(|E|h_{max}^{2}t), respectively, where |E| is the number of edges, h_{max} is the size of the largest local neighborhood of a vertex, and t is the number of iterations required for convergence. We demonstrate the practical efficacy by estimating the spectral density of simulated and real-world undirected and directed graphs.

Top-k Graph Similarity Search Algorithm Based on Chi-Square Statistics in Probabilistic Graphs

Top-k graph similarity search on probabilistic graphs is widely used in various scenarios, such as symptom–disease diagnostics, community discovery, visual pattern recognition, and communication networks. The state-of-the-art method uses the chi-square statistics to speed up the process. The effectiveness of the chi-square statistics solution depends on the effectiveness of the sample observation and expectation. The existing method assumes that the labels in the data graphs are subject to uniform distribution and calculate the chi-square value based on this. In fact, however, the actual distribution of the labels does not meet the requirement of uniform distribution, resulting in a low quality of the returned results. To solve this problem, we propose a top-k similar subgraph search algorithm ChiSSA based on chi-square statistics. We propose two ways to calculate the expectation vector according to the actual distribution of labels in the graph, including the local expectation calculation method based on the vertex neighbors and the global expectation calculation method based on the label distribution of the whole graph. Furthermore, we propose two optimization strategies to improve the accuracy of query results and the efficiency of our algorithm. We conduct rich experiments on real datasets. The experimental results on real datasets show that our algorithm improves the quality and accuracy by an average of 1.66× and 1.68× in terms of time overhead, it improves by an average of 3.41×.

Central limit theorems for local network statistics

Summary Subgraph counts, in particular the number of occurrences of small shapes such as triangles, characterize properties of random networks. As a result, they have seen wide use as network summary statistics. Subgraphs are typically counted globally, making existing approaches unable to describe vertex-specific characteristics. In contrast, rooted subgraphs focus on vertex neighbourhoods, and are fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model that generalizes most statistical network models. This result enables a shift in the statistical analysis of graphs, from estimating network summaries to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that gender and race are significant predictors of local friendship patterns.

Predicting Binding Energies and Electronic Properties of Boron Nitride Fullerenes Using a Graph Convolutional Network.

As isoelectronic counterparts of carbon fullerenes, medium-sized boron nitride clusters also prefer cage structures composed of even-sized polygons. As the cluster size increases, the number of cage isomers grows rapidly, and determining the ground state structure requires a tremendous amount of DFT calculations. Herein, we develop a graph convolutional network (GCN) that can describe the energy of a (BN)n cage by its topology connection. We define a vertex feature vector on a dual polyhedron by the permutation of the neighbor vertices' degree and aggregate the information on vertices by two graph convolutional layers to learn the local feature of the dual polyhedron. The GCN is trained on (BN)28 and subsequently tested on (BN)23 and (BN)24 data sets, which satisfactorily reproduce the order of isomer energies from DFT calculations. We further employ the trained GCN to predict the ground state structures within the size range of n = 25-32, which agree well with DFT results. Using the same GCN framework, we also successfully trained the highest-occupied or lowest-unoccupied orbital energies of (BN)28 isomers. The present graph convolutional network establishes a direct mapping between the topological connection and the energetic or electronic properties of a cage-like cluster or molecule.

Analysing detail preserving capabilities of bilateral, laplacian and taubin mesh filtering methods

Mesh filtering of surfaces is crucial for noise reduction, feature preservation, and mesh simplification in graphics, visualization, and computer vision. In this paper, the detail preservation capacities of 3 frequently used filters, i.e., Bilateral, Laplacian, and Taubin mesh filters, in mesh filtering have been thoroughly examined by experiments conducted on 4 different test meshes. While the Bilateral filter excels in preserving sharp features due to its integration of geometric proximity with intensity similarity, the Laplacian filter prioritizes smoothness by averaging neighboring vertex positions, and the Taubin filter offers a balanced approach by merging attributes of both Laplacian and high-pass filters. The Bilateral filter's primary strength lies in its ability to maintain sharp features on a mesh, ensuring that intricate details are preserved by considering both the spatial closeness and intensity similarity of vertices. The Laplacian filter, although effective in achieving mesh smoothness, has the propensity to excessively smooth out sharp and defining features, potentially causing a loss of critical details in the processed mesh. The Taubin filter integrates the best of both worlds, ensuring smoothness without excessive mesh shrinkage; however, it might not excel in feature preservation as effectively as the Bilateral filter or smooth as uniformly as the Laplacian filter, making it a middle-ground option for certain applications. The statistical analysis of the experimental results has shown that the Taubin method is statistically a more successful mesh filtering method for the test sets used in this paper.

On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws

On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws

IMGA: Efficient In-Memory Graph Convolution Network Aggregation With Data Flow Optimizations

Aggregating features from neighbor vertices is a fundamental operation in Graph Convolution Network (GCN). However, the sparsity in graph data creates poor spatial and temporal locality, causing dynamic and irregular memory access patterns and limiting the performance of aggregation on the Von Neumann architecture. The emerging processing-in-memory (PIM) architecture is based on emerging non-volatile memory (NVM), like Spin-orbit torque Magnetic RAM (SOT-MRAM), and demonstrates promising prospects in alleviating the Von Neumann bottleneck. However, the limited memory capacity of PIM medium still incurs non-negligible data movements between PIM architecture and external memory. To solve this challenge, we propose a SOT-MRAM based in-memory computing architecture, called IMGA, for efficient in-situ graph aggregation. Specifically, we design adaptive data flow management strategies that reuse vertex data in MRAM when processing graphs of different scales and adopt edge data as the control signal source to utilize the graph’s structural information. A reordering optimization strategy leveraging hardware-software co-design principle is proposed to further reduce the costly data movement. Experimental results demonstrate that IMGA achieves an average 2523x and 21x speedup, and 1.03E+6 and 1.04E+3 energy efficiency compared with CPU and GPU, respectively.

Weighted microscopic image reconstruction

Assume we inspect a specimen represented as a collection of points and our task is to learn a physical value associated with each point. However, performing a direct measurement is impossible since it damages the specimen. The alternative is to employ aggregate measuring techniques (e.g., CT or MRI), whereby measurements are taken over subsets of points, and the exact values at each point are subsequently extracted by computational methods. In the Minimum Surgical Probing problem (MSP) the inspected specimen is represented by a graph G and a vector ℓ∈Rn that assigns a value ℓi to each vertex i. An aggregate measurement (called probe) centred at vertex i captures its entire neighbourhood, i.e., the outcome of a probe at i is Pi=∑j∈N(i)∪{i}ℓj where N(i) is the open neighbourhood of vertex i. Bar-Noy et al. (2022) gave a criterion whether the vector ℓ can be recovered from the collection of probes P={Pv∣v∈V(G)} alone. However, there are graphs where the vector ℓ cannot be recovered from P alone. In these cases, we are allowed to use surgical probes. A surgical probe at vertex i returns ℓi. The objective of MSP is to recover ℓ from P and G using as few surgical probes as possible.In this paper, we introduce the Weighted Minimum Surgical Probing (WMSP) problem in which a vertex i may have an aggregation coefficient wi, namely Pi=∑j∈N(i)ℓj+wiℓi. We show that WMSP can be solved in polynomial time. Moreover, we analyse the number of required surgical probes depending on the weight vector w. For any graph, we give two boundaries outside of which no surgical probes are needed to recover the vector ℓ. The boundaries are connected to the (Signless) Laplacian matrix.In addition, we consider the special case where w=0→ and explore the range of possible behaviour of WMSP by determining the number of surgical probes necessary in certain graph families, such as trees and various grid graphs.

On the local metric dimension of generalized wheel graph

Metric dimension and local metric dimension are distance-based tools that find their applications in robot navigation, combinatorial optimization, pattern recognition, image processing, integer programming, and many more. A study based on the selection of the minimum number of vertices in a graph that will separate all the neighboring vertices in terms of distance between the vertices is given by the local metric basis problem. It is well known that the problem of finding the local metric basis of a graph is NP-complete. In this paper, the local metric dimension of double wheel graph [Formula: see text], generalized wheel graph [Formula: see text], and closed helm graph [Formula: see text] had been computed. It has been found that these graphs have unbounded local metric dimensions.

QBER: Quantum-based Entropic Representations for un-attributed graphs

In this paper, we propose a novel framework of computing the Quantum-based Entropic Representations (QBER) for un-attributed graphs, through the Continuous-time Quantum Walk (CTQW). To achieve this, we commence by transforming each original graph into a family of k-level neighborhood graphs, where each k-level neighborhood graph encapsulates the connected information between each vertex and its k-hop neighbor vertices, providing a fine representation to reflect the multi-level topological information for the original global graph structure. To further capture the complicated structural characteristics of the original graph through its neighborhood graphs, we propose to characterize the structure of each neighborhood graph with the Average Mixing Matrix (AMM) of the CTQW, that encapsulates the time-averaged behavior of the CTQW evolved on the neighborhood graph. More specifically, we show how the AMM matrix allows us to compute a Quantum Shannon Entropy for each vertex, and thus compute an entropic signature for each neighborhood graph by measuring the averaged value or the Jensen–Shannon Divergence between the entropies of its vertices. For each original graph, the resulting QBER is defined by gauging how the entropic signat ures vary on its k-level neighborhood graphs with increasing k, reflecting the multi-dimensional entropy-based structure information of the original graph. Experiments on standard graph datasets demonstrate the effectiveness of the proposed QBER approach in terms of the classification accuracies. The proposed approach can significantly outperform state-of-the-art entropic complexity measuring methods, graph kernel methods, as well as graph deep learning methods.