Graph Decomposition Research Articles

Decomposition of <i>P</i><sub>6</sub>-Free Chordal Bipartite Graphs

Canonical decomposition for bipartite graphs, which was introduced by Fouquet, Giakoumakis, and Vanherpe (1999), is a decomposition scheme for bipartite graphs associated with modular decomposition. Weak-bisplit graphs are bipartite graphs totally decomposable (i.e., reducible to single vertices) by canonical decomposition. Canonical decomposition comprises series, parallel, and K + S decomposition. This paper studies a decomposition scheme comprising only parallel and K + S decomposition. We show that bipartite graphs totally decomposable by this decomposition are precisely P6-free chordal bipartite graphs. This characterization indicates that P6-free chordal bipartite graphs can be recognized in linear time using the recognition algorithm for weak-bisplit graphs presented by Giakoumakis and Vanherpe (2003).

The alteration of structural network upon transient association between proteins studied using graph theory.

Proteins such as enzymes perform their function by predominant non-covalent bond interactions between transiently interacting units. There is an impact on the overall structural topology of the protein, albeit transient nature of such interactions, that enable proteins to deactivate or activate. This aspect of the alteration of the structural topology is studied by employing protein structural networks, which are node-edge representative models of protein structure, reported as a robust tool for capturing interactions between residues. Several methods have been optimized to collect meaningful, functionally relevant information by studying alteration of structural networks. In this article, different methods of comparing protein structural networks are employed, along with spectral decomposition of graphs to study the subtle impact of protein-protein interactions. A detailed analysis of the structural network of interacting partners is performed across a dataset of around 900 pairs of bound complexes and corresponding unbound protein structures. The variation in network parameters at, around, and far away from the interface are analyzed. Finally, we present interesting case studies, where an allosteric mechanism of structural impact is understood from communication-path detection methods. The results of this analysis are beneficial in understanding protein stability, for future engineering, and docking studies.

Efficient continuous subgraph matching scheme considering data reuse

As the utilization of graph streams in various applications increases, a continuous subgraph matching scheme is required to search for subgraphs that change in real time. In this paper, we present a novel and effective continuous subgraph matching scheme that leverages indexing and employs distributed processing in graph stream environments. To achieve distributed processing, we employ a query graph decomposition policy based on the degree of nodes, allowing us to manage the decomposed subqueries as an index. By reusing indexing information, we significantly reduce the indexing load, which becomes crucial in scenarios where multiple queries are issued simultaneously. To optimize query allocation in the distributed environment, we introduce a cost model that accurately calculates the indexing load for each server. This ensures a balanced distribution of queries, enhancing overall system performance. To perform distributed processing efficiently in stream environments, the proposed scheme is implemented in Storm. We conduct various performance evaluations to demonstrate the superiority of the proposed scheme.

On matching number, decomposition and representation of well-formed graph

On matching number, decomposition and representation of well-formed graph

Finding Hall Blockers by Matrix Scaling

Given a nonnegative matrix [Formula: see text], the matrix scaling problem asks whether A can be scaled to a doubly stochastic matrix [Formula: see text] for some positive diagonal matrices D1, D2. The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization [Formula: see text] and column-normalization [Formula: see text] alternatively. By this algorithm, A converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with A has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph G, which is identified with the 0, 1-matrix AG. Linial et al. (2000) showed that [Formula: see text] iterations for AG decide whether G has a perfect matching. Here, n is the number of vertices in one of the color classes of G. In this paper, we show an extension of this result. If G has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker—a vertex subset X having neighbors [Formula: see text] with [Formula: see text], which is a certificate of the nonexistence of a perfect matching. Specifically, we show that [Formula: see text] iterations can identify one Hall blocker and that further polynomial iterations can also identify all parametric Hall blockers X of maximizing [Formula: see text] for [Formula: see text]. The former result is based on an interpretation of the Sinkhorn algorithm as alternating minimization for geometric programming. The latter is on an interpretation as alternating minimization for Kullback–Leibler (KL) divergence and on its limiting behavior for a nonscalable matrix. We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage–Mendelsohn decomposition of a bipartite graph. Funding: K. Hayashi was supported by the Japan Society for the Promotion of Science [Grant JP19J22605]. H. Hirai was supported by Precursory Research for Embryonic Science and Technology [Grant JPMJPR192A].

The λ-Fold Spectrum Problem for the Orientations of the Eight-Cycle

A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is K*λKv*, the λ-fold complete symmetric digraph on v vertices, that is, the digraph with λ directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which K*λKv* has a D-decomposition is called the λ-fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The λ-fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the λ-fold spectrum problem for each of the remaining 21 orientations.

A Consensus-Based Alternating Direction Method for Mixed-Integer and PDE-Constrained Gas Transport Problems

We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel and may result in simpler classes of optimization problems because not every block necessarily contains mixed-integer or nonlinear aspects. For achieving feasibility at the interfaces of the several blocks, we employ a tailored consensus-based penalty alternating direction method. Our numerical results show that such decomposition techniques can outperform the baseline approach of just solving the overall MINLP from scratch. However, a complete answer to the question of how to decompose MINLPs on graphs in dependence of the given model is still an open topic for future research. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Grant TRR 154].

Hyper Markov law in undirected graphical models with its applications

By exploring the prime decomposition of undirected graphs, this work investigates the hyper Markov property within the framework of arbitrary undirected graph, which can be seen as the generalization of that for decomposable graphical models proposed by Dawid and Laurizten. The proposed hyper Markov properties of this article can be used to characterize the conditional independence of a distribution or a statistical quantity and helpful to simplify the likelihood ratio functions for statistical test for two different graphs obtained by removing or adding one edge. As an application of these properties, the G-Wishart law is introduced as a prior law for graphical Gaussian models for Bayesian posterior updating, and a hypothesis test for precision matrix is designed to determine the model structures. Our simulation experiments are implemented by using the Gibbs Sampler algorithm and the results show that it performs better in convergence speed.

Supervised spectral feature learning for fine-grained classification in small data set

Fine-grained image classification is a challenging task due to the small inter-class variance, the large intra-class difference, and the small training data. Traditional methods typically rely on large-scale training samples with annotated part annotations, making them costly and severely limiting their application area. In this paper, we propose an effective and weakly supervised fine-grained classification framework. In this framework, a discriminative class-specific spectral feature is learned by intra-class spectral coupling and inter-class spectral decoupling under the weak supervision of image-level category labels, and then the new input images are classified based on the learned class-specific spectral feature. Different from existing strong supervised methods, the proposed technique creatively combines weak supervision of the image-level category labels with unsupervised spectral graph decomposition, not relying on large-scale training samples with dense part annotations, which are heavily labor-consuming. The performance of the proposed methods has been verified on four kinds of typical datasets: the JAFFE dataset, the Yale database, the UCI-CMU face database, and the neural foramina dataset. The satisfactory classification results have been achieved by the proposed method in expression recognition on the JAFFE dataset (with a mean accuracy of 95.31%), face recognition on the Yale database (with a mean accuracy of 98.79%), object recognition on the UCI-CMU face database (with a mean accuracy of 96.96%), and disease grading on the neural foramina dataset (with a mean accuracy of 92.09%). Compared with most state-of-the-art methods, the proposed method has superior classification performance in the small data set.

Multi - Decomposition of Complete Bipartite Graphs into Stars and Bowties of Size l

Multi - Decomposition of Complete Bipartite Graphs into Stars and Bowties of Size l

Scheduling to reduce close contacts: resolvable grid graph decomposition and packing

Scheduling to reduce close contacts: resolvable grid graph decomposition and packing

Unstable graphs and packing into fifth power

In a graph [Formula: see text], a subset [Formula: see text] of the vertex set [Formula: see text] is a module (or interval, clan) of [Formula: see text] if every vertex outside [Formula: see text] is adjacent to all or none of [Formula: see text]. The empty set, the singleton sets, and the full set of vertices are trivial modules. The graph [Formula: see text] is indecomposable (or prime) if all its modules are trivial. If [Formula: see text] is indecomposable, we say that an edge [Formula: see text] of [Formula: see text] is a removable edge if [Formula: see text] is indecomposable (here [Formula: see text]). The graph [Formula: see text] is said to be unstable if it has no removable edges. For a positive integer [Formula: see text], the [Formula: see text]th power [Formula: see text] of a graph [Formula: see text] is the graph obtained from [Formula: see text] by adding an edge between all pairs of vertices of [Formula: see text] with distance at most [Formula: see text]. A graph [Formula: see text] of order [Formula: see text] (i.e., [Formula: see text]) is said to be [Formula: see text]-placeable into [Formula: see text], if [Formula: see text] contains [Formula: see text] edge-disjoint copies of [Formula: see text]. In this paper, we answer a question, suggested by Boudabbous in a personal communication, concerning unstable graphs and packing into their fifth power as follows: First, we give a characterization of the unstable graphs which is deduced from the results given by Ehrenfeucht, Harju and Rozenberg (the theory of [Formula: see text]-structures: a framework for decomposition and transformation of graphs). Second, we prove that every unstable graph [Formula: see text] is [Formula: see text]-placeable into [Formula: see text].

ICIF: Image fusion via information clustering and image features.

Image fusion technology is employed to integrate images collected by utilizing different types of sensors into the same image to generate high-definition images and extract more comprehensive information. However, all available techniques derive the features of the images by utilizing each sensor separately, resulting in poorly correlated image features when different types of sensors are utilized during the fusion process. The fusion strategy to make up for the differences between features alone is an important reason for the poor clarity of fusion results. Therefore, this paper proposes a fusion method via information clustering and image features (ICIF). First, the weighted median filter algorithm is adopted in the spatial domain to realize the clustering of images, which uses the texture features of an infrared image as the weight to influence the clustering results of the visible light image. Then, the image is decomposed into the base layer, bright detail layer, and dark detail layer, which improves the correlations between the layers after conducting the decomposition of a source graph. Finally, the characteristics of the images collected by utilizing sensors and feature information between the image layers are used as the weight reference of the fusion strategy. Hence, the fusion images are reconstructed according to the principle of extended texture details. Experiments on public datasets demonstrate the superiority of the proposed strategy over state-of-the-art methods. The proposed ICIF highlighted targets and abundant details as well. Moreover, we also generalize the proposed ICIF to fuse images with different sensors, e.g., medical images and multi-focus images.

A Heuristic for Direct Product Graph Decomposition

In this paper we describe a heuristic for decomposing a directed graph into factors according to the direct product (also known as Kronecker, cardinal or tensor product). Given a directed, unweighted graph $G$ with adjacency matrix $\mathbf{Adj}(G)$, our heuristic aims at identifying two graphs $G_1$ and $G_2$ such that $G = G_1 \times G_2$, where $G_1 \times G_2$ is the direct product of $G_1$ and $G_2$. For undirected, connected graphs it has been shown that graph decomposition is "at least as difficult" as graph isomorphism; therefore, polynomial-time algorithms for decomposing a general directed graph into factors are unlikely to exist. Although graph factorization is a problem that has been extensively investigated, the heuristic proposed in this paper represents - to the best of our knowledge - the first computational approach for general directed, unweighted graphs. We have implemented our algorithm using the MATLAB environment; we report on a set of experiments that show that the proposed heuristic solves reasonably-sized instances in a few seconds on general-purpose hardware. Although the proposed heuristic is not guaranteed to find a factorization, even if one exists; however, it always succeeds on all the randomly-generated instances used in the experimental evaluation.

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example of a problem that does not behave computationally the same in all subclasses of chordal graphs is the Subset Feedback Vertex Set (SFVS) problem: given a vertex-weighted graph G=(V,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=(V,E)$$\\end{document} and a set S⊆V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\\subseteq V$$\\end{document}, we seek for a vertex set of minimum weight that intersects all cycles containing a vertex of S. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains np-complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider the leafage, which measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on n vertices with leafage ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}, we provide an algorithm for solving SFVS with running time nO(ℓ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{O(\\ell )}$$\\end{document}, thus improving upon nO(ℓ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{O(\\ell ^2)}$$\\end{document}, which is the running time of an approach that utilizes the previously known algorithm for graphs with bounded mim-width. We complement our result by showing that SFVS is w[1]-hard parameterized by ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}. Pushing further our positive result, it is natural to also consider the vertex leafage, which measures the minimum upper bound on the number of leaves of every subtree in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains np-complete on undirected path graphs, i.e., chordal graphs having vertex leafage at most two. Lastly, we provide a polynomial-time algorithm for solving SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs with mim-width one, which is faster than the approach of constructing a graph decomposition of mim-width one and applying the previously known algorithm for graphs with bounded mim-width.

Disentangling the Folklore Hairball

Abstract The ATU tale type index and the Motif Index of Folk-Literature have formed the basis for many comparative folktale studies. While the indices have been used extensively for the study of small groups of folktales and their associated motifs, there have been few attempts of describing a large linguistically and culturally unified corpus through its indexing. The study corpus consists of 2,606 folktales collected by Evald Tang Kristensen in nineteenth century Denmark, which were later indexed according to the second revised edition of the Aarne-Thompson index. We adjust this older index to align with the current ATU index. By creating linked network representations of the ATU index and the MI, as well as updating the Brandt indexing of the Danish folktales, we generate a network with 19,738 nodes and 28,292 edges, where nodes can be ATU numbers, MI numbers, Danish folktales, storytellers, or places of collection. By embedding all the Danish stories in this network, we provide a large-scale overview of the Danish folktale tradition. We introduce two novel interrelated network decomposition methods for the study of folktale collections at corpus scale: fixed points of degree peeling and graph fragments. The resulting analysis of the Danish corpus supports comparison with other traditions. Any collection that is similarly indexed can be embedded in this ATU+MI network and then subjected to the same interrelated graph decompositions.

Convexity Hierarchies in Grid Networks

Several algorithms for path planning in grid networks rely on graph decomposition to reduce the search space size; either by constructing a search data structure on the components, or by using component information for A* guidance. The focus is usually on obtaining components of roughly equal size with few boundary nodes each. In this paper, we consider the problem of splitting a graph into convex components. A convex component is characterized by the property that for all pairs of its members, the shortest path between them is also contained in it. Thus, given a source node, a target node, and a (small) convex component that contains both of them, path planning can be restricted to this component without compromising optimality. We prove that it is NP-hard to find a balanced node separator that splits a given graph into convex components. However, we also present and evaluate heuristics for (hierarchical) convex decomposition of grid networks that perform well across various benchmarks. Moreover, we describe how existing path planning methods can benefit from the computation of convex components. As one main outcome, we show that contraction hierarchies become up to an order of magnitude faster on large grids when the contraction order is derived from a convex graph decomposition.

On Orthogonal Double Covers and Decompositions of Complete Bipartite Graphs by Caterpillar Graphs

Nowadays, graph theory is one of the most exciting fields of mathematics due to the tremendous developments in modern technology, where it is used in many important applications. The orthogonal double cover (ODC) is a branch of graph theory and is considered as a special class of graph decomposition. In this paper, we decompose the complete bipartite graphs Kx,x by caterpillar graphs using the method of ODCs. The article also deals with constructing the ODCs of Kx,x by general symmetric starter vectors of caterpillar graphs such as stars–caterpillar, the disjoint copies of cycles–caterpillars, complete bipartite caterpillar graphs, and the disjoint copies of caterpillar paths. We decompose the complete bipartite graph by the complete bipartite subgraphs and by the disjoint copies of complete bipartite subgraphs using general symmetric starter vectors. The advantage of some of these new results is that they enable us to decompose the giant networks into large groups of small networks with the comprehensive coverage of all parts of the giant network by using the disjoint copies of symmetric starter subgraphs. The use case of applying the described theory for various applications is considered.

Learning Decomposed Spatial Relations for Multi-Variate Time-Series Modeling

Modeling multi-variate time-series (MVTS) data is a long-standing research subject and has found wide applications. Recently, there is a surge of interest in modeling spatial relations between variables as graphs, i.e., first learning one static graph for each dataset and then exploiting the graph structure via graph neural networks. However, as spatial relations may differ substantially across samples, building one static graph for all the samples inherently limits flexibility and severely degrades the performance in practice. To address this issue, we propose a framework for fine-grained modeling and utilization of spatial correlation between variables. By analyzing the statistical properties of real-world datasets, a universal decomposition of spatial correlation graphs is first identified. Specifically, the hidden spatial relations can be decomposed into a prior part, which applies across all the samples, and a dynamic part, which varies between samples, and building different graphs is necessary to model these relations. To better coordinate the learning of the two relational graphs, we propose a min-max learning paradigm that not only regulates the common part of different dynamic graphs but also guarantees spatial distinguishability among samples. The experimental results show that our proposed model outperforms the state-of-the-art baseline methods on both time-series forecasting and time-series point prediction tasks.

Multiscale analysis of pangenomes enables improved representation of genomic diversity for repetitive and clinically relevant genes.

Advancements in sequencing technologies and assembly methods enable the regular production of high-quality genome assemblies characterizing complex regions. However, challenges remain in efficiently interpreting variation at various scales, from smaller tandem repeats to megabase rearrangements, across many human genomes. We present a PanGenome Research Tool Kit (PGR-TK) enabling analyses of complex pangenome structural and haplotype variation at multiple scales. We apply the graph decomposition methods in PGR-TK to the class II major histocompatibility complex demonstrating the importance of the human pangenome for analyzing complicated regions. Moreover, we investigate the Y-chromosome genes, DAZ1/DAZ2/DAZ3/DAZ4, of which structural variants have been linked to male infertility, and X-chromosome genes OPN1LW and OPN1MW linked to eye disorders. We further showcase PGR-TK across 395 complex repetitive medically important genes. This highlights the power of PGR-TK to resolve complex variation in regions of the genome that were previously too complex to analyze.