Hypergraph Research Articles

Optimal Graphs in the Enhanced Mesh Networks

The degree diameter problem explores the biggest graph (in terms of number of nodes) subject to some restrictions on the valency and the diameter of the graph. The restriction on the valency of the graph does not impose any condition on the number of edges (apart from taking the graph simple), so the resulting graph may be thought of as being embedded in the complete graph. In a generality of the said problem, the graph is taken to be embedded in any connected host graph. In this article, host graph is considered as the enhanced mesh network constructed from the grid network. This article provides some exact values for the said problem and also gives some bounds for the optimal graphs.

Co-occurrence patterns in diagnostic data.

SummaryWe demonstrate how graph decomposition techniques can be employed for the visualization of hierarchical co‐occurrence patterns between medical data items. Our research is based on Gaifman graphs (a mathematical concept introduced in Logic), on specific variants of this concept, and on existing graph decomposition notions, specifically, graph modules and the clan decomposition of so‐called 2‐structures. The construction of the Gaifman graphs from a dataset is based on co‐occurrence, or lack of it, of items in the dataset. We may select a discretization on the edge labels to aim at one among several Gaifman graph variants. Then, the decomposition of the graph may provide us with visual information about the data co‐occurrences, after which one can proceed to more traditional statistical analysis.

On the maximum number of copies of H in graphs with given size and order

AbstractWe study the maximum number of copies of a graph in graphs with a given number of vertices and edges. We show that for any fixed graph is asymptotically realized by the quasi‐clique provided that the edge density is sufficiently large. We also investigate a variant of this problem, when the host graph is bipartite.

Minimalist designs

The iterative absorption method has recently led to major progress in the area of (hyper‐)graph decompositions. Among other results, a new proof of the existence conjecture for combinatorial designs, and some generalizations, was obtained. Here, we illustrate the method by investigating triangle decompositions: We give a simple proof that a triangle‐divisible graph of large minimum degree has a triangle decomposition and prove a similar result for quasi‐random host graphs.

Resolution based algorithms for the transversal hypergraph generation problem

We study the problem, given a hypergraph H, how to generate its transversal (or dual) hypergraph G, using tools from Boolean resolution. We show how to compute the residual dual of H given a subset E of its minimal transversals. This is the hypergraph whose minimal transversals are exactly those missing from E to become the dual of H. We give a novel algorithm for the problem based on the notion of subtransversal. Though its complexity is superpolynomial, it gives new insight to the problem of hypergraph duality. Another variant of the algorithm seems also promising to efficiently test hypergraph duality in practice.

Model-Checking on Ordered Structures

We study the model-checking problem for first- and monadic second-order logic on finite relational structures. The problem of verifying whether a formula of these logics is true on a given structure is considered intractable in general, but it does become tractable on interesting classes of structures, such as on classes whose Gaifman graphs have bounded treewidth. In this article, we continue this line of research and study model-checking for first- and monadic second-order logic in the presence of an ordering on the input structure. We do so in two settings: the general ordered case, where the input structures are equipped with a fixed order or successor relation, and the order-invariant case, where the formulas may resort to an ordering, but their truth must be independent of the particular choice of order. In the first setting we show very strong intractability results for most interesting classes of structures. In contrast, in the order-invariant case we obtain tractability results for order-invariant monadic second-order formulas on the same classes of graphs as in the unordered case. For first-order logic, we obtain tractability of successor-invariant formulas on classes whose Gaifman graphs have bounded expansion. Furthermore, we show that model-checking for order-invariant first-order formulas is tractable on coloured posets of bounded width.

Dual Hyper-Graph Regularized Supervised NMF for Selecting Differentially Expressed Genes and Tumor Classification.

Non-negative matrix factorization (NMF) is a dimensionality reduction technique based on high-dimensional mapping. It can learn part-based representations effectively. In this paper, we propose a method called Dual Hyper-graph Regularized Supervised Non-negative Matrix Factorization (HSNMF). To encode the geometric information of the data, the hyper-graph is introduced into the model as a regularization term. The advantage of hyper-graph learning is to find higher order data relationship to enhance data relevance. This method constructs the data hyper-graph and the feature hyper-graph to find the data manifold and the feature manifold simultaneously. The application of hyper-graph theory in cancer datasets can effectively find pathogenic genes. The discrimination information is further introduced into the objective function to obtain more information about the data. Supervised learning with label information greatly improves the classification effect. Furthermore, the real datasets of cancer usually contain sparse noise, so the L2,1-norm is applied to enhance the robustness of HSNMF algorithm. Experiments under The Cancer Genome Atlas (TCGA) datasets verify the feasibility of the HSNMF method.

Induced subgraphs with many repeated degrees

Erdős, Fajtlowicz and Staton asked for the least integer f(k) such that every graph with more than f(k) vertices has an induced regular subgraph with at least k vertices. Here we consider the following relaxed notions. Let g(k) be the least integer such that every graph with more than g(k) vertices has an induced subgraph with at least k repeated degrees and let h(k) be the least integer such that every graph with more than h(k) vertices has an induced subgraph with at least k maximum degree vertices. We obtain polynomial lower bounds for h(k) and g(k) and nontrivial linear upper bounds when the host graph has bounded maximum degree.

The Turán Number of Berge-$K_4$ in 3-Uniform Hypergraphs

For a graph $G=(V,E)$, a hypergraph $H$ is called a Berge-$G$ if there is a bijection $f:E(G)\mapsto E(H)$ such that $e\subseteq f(e)$ for all $e\in E(G)$. The family of Berge-$G$ hypergraphs is de...

Embedding of Complete graphs and Cycles into Grids with holes

Abstract An important feature of an interconnection network is its ability to efficiently simulate one architecture by another. Such a simulation problem can be mathematically formulated as a graph embedding problem. Although the definition of an embedding is an into mapping from Guest Graph to Host Graph, so far in the literature, the embedding has been considered as a mapping from G onto H. In other words, the number of processors in G and H are considered to be the same. In this paper, we increase the number of processors in H by 1. The question is to find the processor in H which does not have the pre-image under the embedding mapping, so that the wirelength of the embedding is minimum. We relate this problem to finding transmission of vertices in the host graph.

A rainbow blow-up lemma for almost optimally bounded edge-colourings

Abstract A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and graph labellings.

Cluster ensemble selection using balanced normalized mutual information

A bad partition in an ensemble will be removed by a cluster ensemble selection framework from the final ensemble. It is the main idea in cluster ensemble selection to remove these partitions (bad partitions) from the selected ensemble. But still, it is likely that one of them contains some reliable clusters. Therefore, it may be reasonable to apply the selection phase on cluster level. To do this, a cluster evaluation metric is needed. Some of these metrics have been recently introduced; each of them has its limitations. The weak points of each method have been addressed in the paper. Subsequently, a new metric for cluster assessment has been introduced. The new measure is named Balanced Normalized Mutual Information (BNMI) criterion. It balances the deficiency of the traditional NMI-based criteria. Additionally, an innovative cluster ensemble approach has been proposed. To create the consensus partition considering the elected clusters, a set of different aggregation-functions (called also consensus-functions) have been utilized: the ones which are based upon the co-association matrix (CAM), the ones which are based on hyper graph partitioning algorithms, and the ones which are based upon intermediate space. The experimental study indicates that the state-of-the-art cluster ensemble methods are outperformed by the proposed cluster ensemble approach.

Noisy Toric code and random-bond Ising model: The error threshold in a dual picture

It is known that noisy topological quantum codes are related to random bond Ising models where the order-disorder phase transition in the classical model is mapped to the error threshold of the corresponding topological code. On the other hand, there is a dual mapping between classical spin models and quantum Calderbank-Shor-Stean (CSS) states where the partition function of a classical model defined on a hypergraph $H$ is written as an inner product of a product state and a CSS state on dual hypergraph $\tilde{H}$. It is then interesting to see what is the interpretation of the classical phase transition in the random bond Ising model within the framework of the above duality and whether such an interpretation has any connection to the error threshold of the corresponding topological CSS code. In this paper, we consider the above duality relation specifically for a two-dimensional random bond Ising model. We show that the order parameter of this classical system is mapped to a coherence order parameter in a noisy Toric code model. In particular, a quantum phase transition from a coherent phase to a non-coherent phase occurs when the initial coherent state is affected by two sequences of bit-flip quantum channels where a quenched disorder is induced by measurement of the errors after the first channel. On the other hand, the above transition is directly related to error threshold of the Toric code model. Accordingly, and since the noisy process can be applied to other topological CSS states, we conclude that the dual correspondence can also provide a useful tool for the study of error thresholds in different topological CSS codes.

Dual Hypergraph Regularized PCA for Biclustering of Tumor Gene Expression Data

Clustering is a powerful approach to analyze gene expression data which is crucial to the investigation of effective treatment of cancer. Many graph regularize-based clustering methods have been proposed and shown to be superior to the traditional clustering methods. However, they only focus on the inner structure in samples and fail to take the feature manifold into account. In gene expression data, it's practical to hypothesize that both the samples and the genes lie on nonlinear low dimensional manifolds, namely sample manifold and gene manifold, respectively. Therefore in this paper, incorporating the geometric structures in both samples and features, we propose a Dual Hypergraph Regularized PCA (DHPCA) method for biclustering of tumor data. First, for gene expression data, we construct two hypergraphs, i.e., sample hypergraph and gene hypergraph, to estimate the intrinsic geometric structures of samples and genes. Then, we introduce the hypergraph regularization on both gene side and sample side. Finally, our biclustering method is formulated as two hypergraph regularized PCA with closed-form solution. We experimentally validate our proposed DHPCA algorithm on real applications and the promising results indicate its potential in high dimension data analysis.

Parallel Semi-Supervised Big Data Clustering Based on Mapreduce Technology

In the area of information technology, a speedy sensational technology is big data. Big data brings tremendous challenges to extract valuable hidden knowledge. Data mining techniques can be used over big data to extract valuable knowledge for decision making. Big data results in high heterogeneity because it consists of various inter-related kinds of objects such as audios, texts, and images. In addition to this, the inter-related kinds of objects carry different information. So, in this paper clustering techniques are introduced to separate objects into several clusters. It also reduces the computational complexity of classifiers. A Possibilistic c-Means (PCM) algorithm was introduced to group the objects in big data. PCM replicated the characteristic of each object to different clusters effectively and it had capability to avoid the corruption of noise in the clustering process. However, PCM is not more efficient for big data and it cannot confine the complex correlation over multiple modalities of the heterogeneous data objects. So, a Parallel Semi-supervised Multi-Ant Colonies Clustering (PSMACC) is introduced for big data clustering. Initially, the PSMACC splits the data into number of partitions and each partition is processed in mappers. Each mapper generates a diverse collection of three clustering components using the semi-supervised ant colony clustering algorithm with various moving speeds. Then, a hyper graph model was used to combine three clustering components. Finally, two constraints such as Must-Link (ML) and Cannot-Link (CL) are included to form a consensus clustering. Finally, the intermediate results of each mapper are combined in the reducer. However, the overhead of iteration in PSMACC is overwhelming which affects the performance of PSMACC. So, a Parallel Semi-supervised Multi-Imperialist Competitive Algorithm (PSMICA) is proposed to cluster the big data. In PSMICA, each mapper processes the ICA where initial population is called countries. Some of the best countries in the population chosen as the imperialists and the remaining countries form the colonies of these imperialists. The colonies move towards the imperialists based on the distance between them. The intermediate results of each mapper are combined in reducer to get the final clustering result.

Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks (THLNs)

It is known widely that an interconnection network can be denoted by a graph G = ( V , E ) , where V denotes the vertex set and E denotes the edge set. Investigating structures of G is necessary to design a suitable topological structure of interconnection network. One of the critical issues in evaluating an interconnection network is graph embedding, which concerns whether a host graph contains a guest graph as its subgraph. Linear arrays (i.e., paths) and rings (i.e., cycles) are two ordinary guest graphs (or basic networks) for parallel and distributed computation. In the process of large-scale interconnection network operation, it is inevitable that various errors may occur at nodes and edges. It is significant to find an embedding of a guest graph into a host graph where all faulty nodes and edges have been removed. This is named as fault-tolerant embedding. The twisted hypercube-like networks ( T H L N s ) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of n-dimensional (n-D) T H L N s . Let G n be an n-D T H L N and F be a subset of V ( G n ) ∪ E ( G n ) with | F | ≤ n - 2 . We show that for two different arbitrary correct vertices u and v, there is a faultless path P u v of every length l with 2 n - 1 - 1 ≤ l ≤ 2 n - f v - 1 - α , where α = 0 if vertices u and v form a normal vertex-pair and α = 1 if vertices u and v form a weak vertex-pair in G n - F ( n ≥ 5 ).

On the Chromatic Number of Matching Kneser Graphs

AbstractIn an earlier paper, the present authors (2015) introduced thealtermatic numberof graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.

Planar anti-Ramsey numbers of paths and cycles

Motivated by anti-Ramsey numbers introduced by Erdős, Simonovits and Sós in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. Given a positive integer n and a planar graph H, let Tn(H) be the family of all plane triangulations T on n vertices such that T contains a subgraph isomorphic to H. The planar anti-Ramsey number ofH, denoted arP(n,H), is the maximum number of colors in an edge-coloring of a plane triangulation T∈Tn(H) such that T contains no rainbow copy of H. Analogously to anti-Ramsey numbers and Turán numbers, planar anti-Ramsey numbers are closely related to planar Turán numbers, where the planar Turán number ofH is the maximum number of edges of a planar graph on n vertices without containing H as a subgraph. The study of arP(n,H) (under the name of rainbow numbers) was initiated by Horňák et al. (2015). In this paper we study planar anti-Ramsey numbers for paths and cycles. We first improve existing lower bound for arP(n,Ck) when k≥5 and n≥k2−k. Then, using the main ideas in the above-mentioned paper, we obtain upper bounds for arP(n,C6) when n≥8 and arP(n,C7) when n≥13, respectively. Finally, we establish lower bounds for arP(n,Pk) when n≥k≥8.

Algebraic graph rewriting with controlled embedding

Graph transformation is a specification technique suitable for a wide range of applications, specially the ones that require a sophisticated notion of state. In graph transformation, states are represented by graphs and actions are specified by rules. Most algebraic approaches to graph transformation proposed in the literature ensure that if an item is preserved by a rule, so are its connections with the graph where it is embedded. But there are applications in which it is desirable to specify different embeddings. For example when cloning an item, there may be a need to handle the original and the copy in different ways. We propose a new algebraic approach to graph transformation, AGREE: Algebraic Graph Rewriting with controllEd Embedding, where rules allow one to specify how the embedding should be carried out. We define this approach in the framework of classified categories which are categories endowed with partial map classifiers. This new approach leads to graph transformations in which effects may be non-local, e.g. a rewrite step may alter a node of the host graph which is outside the image of the left-hand side of the considered rule. We propose a syntactic condition on AGREE rules which guarantees the locality of transformations. We also compare AGREE with other algebraic approaches to graph transformation.

Anti-Ramsey Problems in Complete Bipartite Graphs for t Edge-Disjoint Rainbow Spanning Subgraphs: Cycles and Matchings

A subgraph H of an edge-colored graph G is rainbow if all of its edges have different colors. The anti-Ramsey number is the maximum number of colors in an edge-coloring of G with no rainbow copy of H. Originally a complete graph was considered as G. In this paper, we consider a complete bipartite graph as the host graph and discuss some results for the graph H being hamiltonian cycle and perfect matching. Let $$c(K_{p,p},t)$$ and $$m(K_{p,p},t)$$ be the maximum number of colors in an edge-coloring of the complete bipartite graph $$K_{p,p}$$ not having t edge-disjoint rainbow hamiltonian cycles and perfect matchings, respectively. We prove that $$c(K_{p,p},t)=p^2-p+t$$ for $$t \ge 2$$ , $$p\ge 4t-1$$ and $$m(K_{p,p},t)=p^2-2p+t+1$$ for $$t \ge $$ 2, $$p\ge 2t+8$$ .