Neighborhood Hypergraph Research Articles

Algebraic implications of neighborhood hypergraphs and their transversal hypergraphs

In this paper, we unfold balanced and totally balanced neighborhood hypergraphs to discover new classes of normally torsion-free monomial ideals. As a consequence, we establish that the closed neighborhood ideals and the dominating ideals of strongly chordal graphs are normally torsion-free. We discuss the stable sets of associated primes of the dominating ideals of cycles and characterize all the cycles with normally torsion-free dominating ideals.

A Dual Neighborhood Hypergraph Neural Network for Change Detection in VHR Remote Sensing Images

The very high spatial resolution (VHR) remote sensing images have been an extremely valuable source for monitoring changes occurring on the Earth’s surface. However, precisely detecting relevant changes in VHR images still remains a challenge, due to the complexity of the relationships among ground objects. To address this limitation, a dual neighborhood hypergraph neural network is proposed in this article, which combines multiscale superpixel segmentation and hypergraph convolution to model and exploit the complex relationships. First, the bi-temporal image pairs are segmented under two scales and fed to a pre-trained U-net to obtain node features by treating each object under the fine scale as a node. The dual neighborhood is then defined using the father-child and adjacent relationships of the segmented objects to construct the hypergraph, which permits models to represent higher-order structured information far more complex than the conventional pairwise relationships. The hypergraph convolutions are conducted on the constructed hypergraph to propagate the label information from a small amount of labeled nodes to the other unlabeled ones by the node-edge-node transformation. Moreover, to alleviate the problem of imbalanced sampling, the focal loss function is adopted to train the hypergraph neural network. The experimental results on optical, SAR and heterogeneous optical/SAR data sets demonstrate that the proposed method offersbetter effectiveness and robustness compared to many state-of-the-art methods.

Decision-making methods based on fuzzy soft competition hypergraphs

Fuzzy soft set theory is an effective framework that is utilized to determine the uncertainty and plays a major role to identify vague objects in a parametric manner. The existing methods to discuss the competitive relations among objects have some limitations due to the existence of different types of uncertainties in a single mathematical structure. In this research article, we define a novel framework of fuzzy soft hypergraphs that export the qualities of fuzzy soft sets to hypergraphs. The effectiveness of competition methods is enhanced with the novel notion of fuzzy soft competition hypergraphs. We study certain types of fuzzy soft competition hypergraphs to illustrate different relations in a directed fuzzy soft network using the concepts of height, depth, union, and intersection simultaneously. We introduce the notions of fuzzy soft k-competition hypergraphs and fuzzy soft neighborhood hypergraphs. We design certain algorithms to compute the strength of competition in fuzzy soft directed graphs that reduce the calculation complexity of existing fuzzy-based non-parameterized models. We analyze the significance of our proposed theory with a decision-making problem. Finally, we present graphical, numerical, as well as theoretical comparison analysis with existing methods that endorse the applicability and advantages of our proposed approach.

Neighborhood hypergraph model for topological data analysis

Abstract Hypergraph, as a generalization of the notions of graph and simplicial complex, has gained a lot of attention in many fields. It is a relatively new mathematical model to describe the high-dimensional structure and geometric shapes of data sets. In this paper,we introduce the neighborhood hypergraph model for graphs and combine the neighborhood hypergraph model with the persistent (embedded) homology of hypergraphs. Given a graph,we can obtain a neighborhood complex introduced by L. Lovász and a filtration of hypergraphs parameterized by aweight function on the power set of the vertex set of the graph. Theweight function can be obtained by the construction fromthe geometric structure of graphs or theweights on the vertices of the graph. We show the persistent theory of such filtrations of hypergraphs. One typical application of the persistent neighborhood hypergraph is to distinguish the planar square structure of cisplatin and transplatin. Another application of persistent neighborhood hypergraph is to describe the structure of small fullerenes such as C20. The bond length and the number of adjacent carbon atoms of a carbon atom can be derived from the persistence diagram. Moreover, our method gives a highly matched stability prediction (with a correlation coefficient 0.9976) of small fullerene molecules.

An algorithm to compute the strength of competing interactions in the Bering Sea based on pythagorean fuzzy hypergraphs

The networks of various problems have competing constituents, and there is a concern to compute the strength of competition among these entities. Competition hypergraphs capture all groups of predators that are competing in a community through their hyperedges. This paper reintroduces competition hypergraphs in the context of Pythagorean fuzzy set theory, thereby producing Pythagorean fuzzy competition hypergraphs. The data of real-world ecological systems posses uncertainty, and the proposed hypergraphs can efficiently deal with such information to model wide range of competing interactions. We suggest several extensions of Pythagorean fuzzy competition hypergraphs, including Pythagorean fuzzy economic competition hypergraphs, Pythagorean fuzzy row as well as column hypergraphs, Pythagorean fuzzy k-competition hypergraphs, m-step Pythagorean fuzzy competition hypergraphs and Pythagorean fuzzy neighborhood hypergraphs. The proposed graphical structures are good tools to measure the strength of direct and indirect competing and non-competing interactions. Their aptness is illustrated through examples, and results support their intrinsic interest. We propose algorithms that help to compose some of the presented graphical structures. We consider predator-prey interactions among organisms of the Bering Sea as an application: Pythagorean fuzzy competition hypergraphs encapsulate the competing relationships among its inhabitants. Specifically, the algorithm which constructs the Pythagorean fuzzy competition hypergraphs can also compute the strength of competing and non-competing relations of this scenario.

A framework of decision making based on bipolar fuzzy competition hypergraphs

Bipolarity plays a key role in different domains such as technology, social networking and biological sciences for illustrating real-world phenomenon using bipolar fuzzy models. In this article, novel concepts of bipolar fuzzy competition hypergraphs are introduced and discuss the application of the proposed model. The main contribution is to illustrate different methods for the construction of bipolar fuzzy competition hypergraphs and their variants. Authors study various new concepts including bipolar fuzzy row hypergraphs, bipolar fuzzy column hypergraphs, bipolar fuzzy k-competition hypergraphs, bipolar fuzzy neighborhood hypergraphs and strong hyperedges. Besides, we develop some relations between bipolar fuzzy k-competition hypergraphs and bipolar fuzzy neighborhood hypergraphs. Moreover, authors design an algorithm to compute the strength of competition among companies in business market. A comparative analysis of the proposed model is discuss with the existing models such bipolar fuzzy competition graphs and fuzzy competition hypergraphs.

Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs

AbstractA hypergraph is said to be 1‐Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of ‐Sperner hypergraphs to graphs. First, we consider several ways of associating hypergraphs to graphs, namely, vertex cover, clique, independent set, dominating set, and closed neighborhood hypergraphs. For each of them, we characterize graphs yielding ‐Sperner hypergraphs. These results give new characterizations of threshold and domishold graphs. Second, we apply a characterization of ‐Sperner hypergraphs to derive decomposition theorems for two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs. These decomposition theorems, based on certain matrix partitions, lead to new classes of graphs of bounded clique‐width and new polynomially solvable cases of three basic domination problems: domination, total domination, and connected domination.

Parameterized and approximation complexity of Partial VC Dimension

We introduce the problem Partial VC Dimension that asks, given a hypergraph H=(X,E) and integers k and ℓ, whether one can select a set C⊆X of k vertices of H such that the set {e∩C,e∈E} of distinct hyperedge-intersections with C has size at least ℓ. The sets e∩C define equivalence classes over E. Partial VC Dimension is a generalization of VC Dimension, which corresponds to the case ℓ=2k, and of Distinguishing Transversal, which corresponds to the case ℓ=|E| (the latter is also known as Test Cover in the dual hypergraph). We also introduce the associated fixed-cardinality maximization problem Max Partial VC Dimension that aims at maximizing the number of equivalence classes induced by a solution set of k vertices. We study the algorithmic complexity of Partial VC Dimension and Max Partial VC Dimension both on general hypergraphs and on more restricted instances, in particular, neighborhood hypergraphs of graphs.

Hypergraph-based Algorithm for Segmentation of Weather Satellite Imagery

Objective: Classification of cloud images through segmentation of automated satellite images to improvise the level of accuracy. Method Analysis: To classify cloud images the hyper graph model uses the idea of maximally bonded subsets that is endowed with integer valued metric are applied to receive the classifications. The widely used hyper graph model is Intensity Neighborhood Hyper graph (INHG) and representation model in this article is Intensity Interval Hyper graph (IIHG). Findings: The results obtained through this process is proved to be more accurate and the time complexity is O(n) in weather prediction. Similarly, the results received through IIHG, which also provides the same computational complexity where all the pixels to be processed with less time. Enhancement: The proposed methodology increases the accuracy level of prediction with less computation time and this work can be enhanced by including pattern recognition in automated processing.

Circular-arc hypergraphs: Rigidity via connectedness

A circular-arc hypergraphH is a hypergraph admitting an arc ordering, that is, a circular ordering of the vertex set V(H) such that every hyperedge is an arc of consecutive vertices. We give a criterion for the uniqueness of an arc ordering in terms of connectedness properties of H. This generalizes the relationship between rigidity and connectedness disclosed by Chen and Yesha (1991) in the case of interval hypergraphs. Moreover, we state sufficient conditions for the uniqueness of tight arc orderings where, for any two hyperedges A and B such that A⊆B≠V(H), the corresponding arcs must share a common endpoint. We notice that these conditions are obeyed for the closed neighborhood hypergraphs of proper circular-arc graphs, implying for them the known rigidity results that were originally obtained using the theory of local tournament graph orientations.

Solving the canonical representation and Star System Problems for proper circular-arc graphs in logspace

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs. This class of hypergraphs corresponds to matrices with the circular ones property, which play an important role in computational genomics. Our results imply that there is a logspace algorithm that decides whether a given matrix has this property.Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs.Note that solving a problem in logspace implies that it is solvable by a parallel algorithm of the class AC1. For the problems under consideration, at most AC2 algorithms were known earlier.

VC-dimension and Erdős–Pósa property

Let G=(V,E) be a graph. A k-neighborhood in G is a set of vertices consisting of all the vertices at distance at most k from some vertex of G. The hypergraph on vertex set V whose edge set consists of all the k-neighborhoods of G for all k is the neighborhood hypergraph of G. Our goal in this paper is to investigate the complexity of a graph in terms of its neighborhoods. Precisely, we define the distance VC-dimension of a graph G as the maximum taken over all induced subgraphs G′ of G of the VC-dimension of the neighborhood hypergraph of G′. For a class of graphs, having bounded distance VC-dimension both generalizes minor closed classes and graphs with bounded clique-width.Our motivation is a result of Chepoi et al. (2007) asserting that every planar graph of diameter 2ℓ can be covered by a bounded number of balls of radius ℓ. In fact, they obtained the existence of a function f such that every set F of balls of radius ℓ in a planar graph admits a hitting set of size f(ν) where ν is the maximum number of pairwise disjoint elements of F.Our goal is to generalize the proof of Chepoi et al. (2007) with the unique assumption of bounded distance VC-dimension of neighborhoods. In other words, the set of balls of fixed radius in a graph with bounded distance VC-dimension has the Erdős–Pósa property.

Neighborhood Hypergraph Based Classification Algorithm for Incomplete Information System

The problem of classification in incomplete information system is a hot issue in intelligent information processing. Hypergraph is a new intelligent method for machine learning. However, it is hard to process the incomplete information system by the traditional hypergraph, which is due to two reasons: (1) the hyperedges are generated randomly in traditional hypergraph model; (2) the existing methods are unsuitable to deal with incomplete information system, for the sake of missing values in incomplete information system. In this paper, we propose a novel classification algorithm for incomplete information system based on hypergraph model and rough set theory. Firstly, we initialize the hypergraph. Second, we classify the training set by neighborhood hypergraph. Third, under the guidance of rough set, we replace the poor hyperedges. After that, we can obtain a good classifier. The proposed approach is tested on 15 data sets from UCI machine learning repository. Furthermore, it is compared with some existing methods, such as C4.5, SVM, NavieBayes, andKNN. The experimental results show that the proposed algorithm has better performance via Precision, Recall, AUC, andF-measure.

Proposed Image Similarity Measurement Model based on Hypergraph

In this article, we propose a new image similarity measurement model (SMM) based hypergraph which is easy to calculate and applicable to various image processing application. Hypergraphs are now used in many domains such as chemistry, engineering and image processing. We present an overview of a hypergraph-based Image representation and the Image Adaptive Neighborhood Hypergraph (IANH). With the IANH it is possible to build a new powerful similarity measurement model. Although the new model is mathematically defined and no human visual system model is explicitly employed, our experiments on various image distortion types indicate the efficient of proposed model.

A novel algorithm for imbalance data classification based on neighborhood hypergraph.

The classification problem for imbalance data is paid more attention to. So far, many significant methods are proposed and applied to many fields. But more efficient methods are needed still. Hypergraph may not be powerful enough to deal with the data in boundary region, although it is an efficient tool to knowledge discovery. In this paper, the neighborhood hypergraph is presented, combining rough set theory and hypergraph. After that, a novel classification algorithm for imbalance data based on neighborhood hypergraph is developed, which is composed of three steps: initialization of hyperedge, classification of training data set, and substitution of hyperedge. After conducting an experiment of 10-fold cross validation on 18 data sets, the proposed algorithm has higher average accuracy than others.

On the total k-domination number of graphs

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |NG[v]∩S| ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V , |NG(v)∩S| ≥ k. The k-transversal number τk(H) of a hypergraph H is the minimum size of a subset S ⊆ V (H) such that |S ∩ e| ≥ k for every edge e ∈ E(H). We know that for any graph G of order n with minimum degree at least k, γ×k(G) ≤ γ×k,t(G) ≤ n. Obviously for every k-regular graph, the upper bound n is sharp. Here, we give a sufficient condition forγ×k,t(G) < n. Then we characterize complete multipartite graphs G with γ×k(G) = γ×k,t(G). We also state that the total k-domination number of a graph is the k-transversal number of its open neighborhood hypergraph, and also the domination number of a graph is the transversal number of its closed neighborhood hypergraph. Finally, we give an upper bound for the total k-domination number of the cross product graph G × H of two graphs G and H in terms on the similar numbers of G and H. Also, we show that this upper bound is strict for some graphs, when k = 1.

A hypergraph-based algorithm for image restoration from salt and pepper noise

An algorithm is designed for the hypergraph (HG) representation of an image, subsequent detection of Salt and Pepper (SP) noise in the image and finally the restoration of the image from this noise. The image is first represented as the set union of hyperedges. As for the hyperedges themselves, these are determined by two Image Neighborhood Hypergraph (INHG) parameters, with the concepts of 8-bit neighborhood and INHG of a graph being central. The images taken up for experimental analyses are subjected to the Contra Harmonic Mean (CHM) filter for SP noise removal. The proposed algorithm exhibits superiority over traditional algorithms and recently proposed ones in terms of visual quality, Peak Signal to Noise Ratio (PSNR) and Mean Absolute Error (MAE). This superior performance of the CHM Filter is solely due to the HG representation of the test images.

Neighborhood hypergraphs of digraphs and some matrix permutation problems

Given a digraph D , the set of all pairs ( N − ( v ) , N + ( v ) ) constitutes the neighborhood dihypergraph N ( D ) of D . The Digraph Realization Problem asks whether a given dihypergraph H coincides with N ( D ) for some digraph D . This problem was introduced by Aigner and Triesch [M. Aigner, E. Triesch, Reconstructing a graph from its neighborhood lists, Combin. Probab. Comput. 2 (2) (1993) 103–113] as a natural generalization of the Open Neighborhood Realization Problem for undirected graphs, which is known to be NP-complete. We show that the Digraph Realization Problem remains NP-complete for orgraphs (orientations of undirected graphs). As a corollary, we show that the Matrix Skew-Symmetrization Problem for square { 0 , 1 , − 1 } matrices ( a i j = − a j i ) is NP-complete. This result can be compared with the known fact that the Matrix Symmetrization Problem for square 0–1 matrices ( a i j = a j i ) is NP-complete. Extending a negative result of Fomin, Kratochvíl, Lokshtanov, Mancini, and Telle [F.V. Fomin, J. Kratochvíl, D. Lokshtanov, F. Mancini, J.A. Telle, On the complexity of reconstructing H -free graphs from their star systems, Manuscript (2007) 11 pp] we show that the Digraph Realization Problem remains NP-complete for almost all hereditary classes of digraphs defined by a unique minimal forbidden subdigraph. Finally, we consider the Matrix Complementation Problem for rectangular 0–1 matrices, and prove that it is polynomial-time equivalent to graph isomorphism. A related known result is that the Matrix Transposability Problem is polynomial-time equivalent to graph isomorphism.

Hypergraph Cuts &amp; Unsupervised Representation for Image Segmentation

This paper presents a novel approach to image segmentation based on hypergraph cut techniques. Natural images contain more components: Edge, homogeneous region, noise. So, to facilitate the natural image analysis, we introduce an Image Neighborhood Hypergraph representation (INH). This representation extracts all features and their consistencies in the image data and its mode of use is close to the perceptual grouping. Then, we formulate an image segmentation problem as a hypergraph partitioning problem and we use the recent k-way hypergraph techniques to find the partitions of the image into regions of coherent brightness/color. Experimental results of image segmentation on a wide range of images from Berkeley Database show that the proposed method provides a significant performance improvement compared with the stat-of-the-art graph partitioning strategy based on Normalized Cut (Ncut) criteria.

Neighborhood hypergraphs of bipartite graphs

AbstractMatrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity and relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular, we recall results from the literature showing that matrix symmetrization is in fact NP‐hard; furthermore, it is equivalent with the problem of recognizing whether a hypergraph can be realized as the neighborhood hypergraph of a graph. There are several variants of the latter problem corresponding to the concepts of open, closed, or mixed neighborhoods. While all these variants are NP‐hard in general, one of them restricted to the bipartite graphs is known to be equivalent with graph isomorphism. Extending this result, we consider several other variants of the bipartite neighborhood recognition problem and show that they all are either polynomial‐time solvable, or equivalent with graph isomorphism. Also, we study uniqueness of neighborhood realizations of hypergraphs and show that, in general, for all variants of the problem, a realization may be not unique. However, we prove uniqueness in two special cases: for the open and closed neighborhood hypergraphs of the bipartite graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 69–95, 2008