Number Of Maximal Cliques Research Articles

Finding Ultrametricity in Data using Topology

The topological ultrametricity index can be approximated by the expected survival time of a dataset in the state of being ultrametric while only distances up to a given value are considered. It is observed that the quotient of the number of connected components by the number of maximal cliques in the Vietoris-Rips graph initially is the survival function of a Weibull distribution. This is shown for some codings of Fisher's Iris data as well as for random samples in the Euclidean hypercube.

Tractable classes for CSPs of arbitrary arity: from theory to practice

The research of this thesis focuses on the analysis of polynomial classes and their practical exploitation for solving constraint satisfaction problems (CSPs) with finite domains. In particular, I worked on bridging the gap between theoretical works and practical results in constraint solvers. Specifically, the goal of this thesis is to find explanation for the effectiveness of solvers, and also to show that studied tractable classes are not artificial since several real-problems among the ones used in the CSP 2008 Competition belong to them. Our work is organized into three main parts. In the first part, we proposed several types of microstructures for CSPs of arbitrary arity which are based on some knwon binary encoding of non-binary CSPs like, dual encoding, hidden-variable transformation and mixed (or double) encoding. These theoretical tools are designed to facilitate the study of tractable classes, sets of CSP instances which can be solved in polytime, when the constraints are non-binary. After that, we propose a new tractable classes of CSPs whose the highlighting should allow on the one hand to explain the effectiveness of solvers of the state of the art namely FC, MAC, RFL and on the second hand to provide the opportunities for easy integration in these solvers. These would include the definition of new tractable classes without using of an ad hoc algorithms as in the traditional case. These new tractable classes are related to the number of maximal cliques in the microstructure of binary or non-binary CSP. In the last part, we focus on the presence of instances belonging to polynomial classes in classical benchmarks used by the CP community. We study in particular the Broken-Triangle Property (BTP) and its extension DBTP to CSP of arbitrary arity. Next, we prove that BTP can also be used to reduce the size of the search space by merging pairs of values on which no broken triangle exists. Finally, we introduce a formal framework, called transformation, and we develop the concept of hidden tractable class that we exploit from an experimental point of view.

The \(m\)-Clique Load and the \(m\)-Clique Sequence of Graphs

This paper introduces the concepts of the \(m\)-clique load, the \(m\)-clique sequence and the \(m\)-clique density of graphs. The number of distinct maximum cliques over all maximal cliques is called the \(m\)-clique load of \(G\) and denoted, \(\diamond(G)\). The \(m\)-clique sequence denoted, \(\diamond\)-sequence of a graph \(G\) with \(\epsilon(G)\ge 1\) is the sequence with entries representing the number of maximal cliques of same order found in \(G\), in descending order. A finite sequence of positive integers each indexed with a distinct positive integer subscript which is \(c\)-graphical, is characterised. The \(m\)-clique density of a graph \(G\) denoted, \(p_{c_i}(G)\) is the probability of uniformly at random, choosing a maximal clique \(K_{c_i}\), \(1\le c_i\le \nu(G)\). Introductory results for certain graph classes and power graphs of balanced caterpillars, \(C^{\cal L}_{P_n}\) are also presented.

Distributed Maximal Clique Computation and Management

Maximal cliques are elementary substructures in a graph and instrumental in graph analysis such as the structural analysis of many complex networks, graph clustering and community detection, network hierarchy detection, emerging pattern mining, vertex importance measures, etc. However, the number of maximal cliques is also notoriously large even for many small real world graphs. This size problem gives rise to challenges in both computing and managing the set of maximal cliques. Many algorithms for computing maximal cliques have been proposed in the literature; however, most of them are sequential algorithms that cannot scale due to the high complexity of the problem, while existing parallel algorithms for computing maximal cliques are mostly immature and especially suffer from skewed workload. As for managing the set of maximal cliques, which is essential due to its large size, there is barely any efficient method for querying or updating the set of maximal cliques. In this paper, we first propose a distributed algorithm built on a share-nothing architecture for computing the set of maximal cliques. We effectively address the problem of skewed workload distribution due to high-degree vertices, which also leads to drastically reduced worst-case time complexity for computing maximal cliques in common real-world graphs. Then, we propose a set of fundamental query operations and efficient algorithms to process the queries, to aid more efficient and effective analysis of the set of maximal cliques. Finally, we also devise algorithms to support efficient update maintenance of the set of maximal cliques when the underlying graph is updated. We verify the efficiency of our algorithms for computing, querying, and updating the set of maximal cliques with a range of real-world graphs from different application domains.

On Bounding the Difference of the Maximum Degree and the Clique Number

For every \(k \in {\mathbb {N}}_0\), we consider graphs in which for any induced subgraph, \(\Delta \le \omega - 1 + k\) holds, where \(\Delta \) is the maximum degree and \(\omega \) is the maximum clique number of the subgraph. We give a finite forbidden induced subgraph characterization for every \(k\). As an application, we find some results on the chromatic number \(\chi \) of a graph. B. Reed stated the conjecture that for every graph, \(\chi \le \lceil \frac{\Delta + \omega + 1 }{2}\rceil \) holds. Since this inequality is fulfilled by graphs in which \(\Delta \le \omega + 2\) holds, our results provide a hereditary graph class for which the conjecture holds.

On Graph-Lagrangians of Hypergraphs Containing Dense Subgraphs

Motzkin and Straus established a remarkable connection between the maximum clique and the Graph-Lagrangian of a graph in (Can. J. Math. 17:533---540, 1965). This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we provide upper bounds on the Graph-Lagrangian of a hypergraph containing dense subgraphs when the number of edges of the hypergraph is in certain ranges. These results support a pair of conjectures introduced by Peng and Zhao (Graphs Comb. 29:681---694, 2013) and extend a result of Talbot (Comb. Probab. Comput. 11:199---216, 2002).

On Lagrangians of r-uniform hypergraphs

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in Can J Math 17:533---540 (1965). This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It has been also applied in spectral graph theory. Estimating the Lagrangians of hypergraphs has been successfully applied in the course of studying the Turan densities of several hypergraphs as well. It is useful in practice if Motzkin---Straus type results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus' result to hypergraphs is false. We attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. In this paper, we give some Motzkin---Straus type results for r-uniform hypergraphs. These results generalize and refine a result of Talbot in Comb Probab Comput 11:199---216 (2002) and a result in Peng and Zhao (Graphs Comb, 29:681---694, 2013).

Coloring the complex networks and its application for immunization strategy

Structural analysis of complex networks has gained more and more concerns, but not enough attention has been paid to the coloring problem in complex networks. In order to understand the relationship between network structure and coloring problem, we investigate the effects of WS, BA networks and different macro-scale parameters on the K-proper coloring. We find that the maximum clique number can generally reflect the trend of K value change, the average degree and the degree correlation have a greater impact on the K value than the heterogeneity and the clustering coefficient. These results are verified on some real-world networks. After coloring the complex networks properly, the independent sets of networks can be obtained. According to the characteristic that any two vertices are not connected in an independent set, we propose a random immunization strategy based on the independent set. Compared with the random immunization, the proposed strategy can make the network more vulnerable, and thus effectively mitigate epidemic spreading. This immunization strategy is simple and practical, which helps to design more efficient immunization strategy.

On the maximum number of cliques in a graph embedded in a surface

This paper studies the following question: given a surface Σ and an integer n, what is the maximum number of cliques in an n-vertex graph embeddable in Σ? We characterise the extremal graphs for this question, and prove that the answer is between 8(n−ω)+2ω and 8n+522ω+o(2ω), where ω is the maximum integer such that the complete graph Kω embeds in Σ. For the surfaces S0, S1, S2, N1, N2, N3 and N4 we establish an exact answer.

Inferring gene coexpression networks for low dose ionizing radiation using graph theoretical algorithms and systems genetics

Materials and methods We have developed a tool chain using novel graph algorithms to extract gene coexpression networks from microarray data. We highlight implementation of our tool chain to investigate the effects of in vivo low dose ionizing radiation treatments on mice. We are using systems genetics approach to investigate the biological effects of low dose (10 cGy) ionizing radiation. We measured the base line gene expression profile from spleen tissue of BXD recombinant inbred mice using Illumina microarrays. The data was filtered using coefficient of variance after robust spline normalization and variance stabilizing transformation. A graph was then derived from this data, with probes as vertices and edges between them representing correlations. The graph was analyzed using our toolkit to find the size and number of maximal cliques. We deployed another tool called paraclique that relaxes clique’s requirement that every edge be present between all vertices. Paraclique enables us to account for inherent noise in the microarray data and stochastic nature of biological processes. Using immunophenotype data from the baseline BXD mice, we employed biclique analysis to determine interactions between genotypes and immunophenotypes (%CD4, %CD3, LN T:B, %CD8, and LN CD4:CD8). We also extracted eQTLs from BXD data using QTL-Reaper from base line gene expression profiles. 1881 transcripts were associated with 686 loci. The eQTLs were classified as cis or trans according to their genomic positions. Besides population level studies we also investigated the differential effect of low dose and high dose (1Gy) of ionizing radiations on spleen gene expression in inbred parental strains (C57BL/6J and DBA/2J) of BXD recombinant inbred mice as well as BALB/c mice, a known radiation-sensitive strain.

Partial characterizations of clique-perfect and coordinated graphs: Superclasses of triangle-free graphs

A graph G is clique-perfect if the cardinality of a maximum clique-independent set of H equals the cardinality of a minimum clique-transversal of H , for every induced subgraph H of G . A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color equals the maximum number of cliques of H with a common vertex, for every induced subgraph H of G . Coordinated graphs are a subclass of perfect graphs. The complete lists of minimal forbidden induced subgraphs for the classes of clique-perfect and coordinated graphs are not known, but some partial characterizations have been obtained. In this paper, we characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph is either paw-free or {gem, W 4 , bull}-free, both superclasses of triangle-free graphs.

Comparison of threshold selection methods for microarray gene co-expression matrices

BackgroundNetwork and clustering analyses of microarray co-expression correlation data often require application of a threshold to discard small correlations, thus reducing computational demands and decreasing the number of uninformative correlations. This study investigated threshold selection in the context of combinatorial network analysis of transcriptome data.FindingsSix conceptually diverse methods - based on number of maximal cliques, correlation of control spots with expressed genes, top 1% of correlations, spectral graph clustering, Bonferroni correction of p-values, and statistical power - were used to estimate a correlation threshold for three time-series microarray datasets. The validity of thresholds was tested by comparison to thresholds derived from Gene Ontology information. Stability and reliability of the best methods were evaluated with block bootstrapping.Two threshold methods, number of maximal cliques and spectral graph, used information in the correlation matrix structure and performed well in terms of stability. Comparison to Gene Ontology found thresholds from number of maximal cliques extracted from a co-expression matrix were the most biologically valid. Approaches to improve both methods were suggested.ConclusionThreshold selection approaches based on network structure of gene relationships gave thresholds with greater relevance to curated biological relationships than approaches based on statistical pair-wise relationships.

Fixed-Parameter Algorithms for Cluster Vertex Deletion

We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.

Partial characterizations of coordinated graphs: line graphs and complements of forests

A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The list of minimal forbidden induced subgraphs for the class of coordinated graphs is not known. In this paper, we present a partial result in this direction, that is, we characterize coordinated graphs by minimal forbidden induced subgraphs when the graph is either a line graph, or the complement of a forest.

NP-hardness of the recognition of coordinated graphs

A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. In previous works, polynomial time algorithms were found for recognizing coordinated graphs within some classes of graphs. In this paper we prove that the recognition problem for coordinated graphs is NP-hard, and it is NP-complete even when restricted to the class of {gem, C4, odd hole}-free graphs with maximum degree four, maximum clique size three and at most three cliques sharing a common vertex.

Partial characterizations of clique-perfect and coordinated graphs: superclasses of triangle-free graphs

Abstract A graph is clique-perfect if the cardinality of a maximum clique-independent set equals the cardinality of a minimum clique-transversal, for all its induced subgraphs. A graph G is coordinated if the chromatic number of the clique graph of H equals the maximum number of cliques of H with a common vertex, for every induced subgraph H of G . Coordinated graphs are a subclass of perfect graphs. The complete lists of minimal forbidden induced subgraphs for the classes of cliqueperfect and coordinated graphs are not known, but some partial characterizations have been obtained. In this paper, we characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph is either paw-free or {gem,W 4 ,bull}-free, two superclasses of triangle-free graphs.

On the Maximum Number of Cliques in a Graph

A \emph{clique} is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with $n$ vertices and $m$ edges; (2) graphs with $n$ vertices, $m$ edges, and maximum degree $\Delta$; (3) $d$-degenerate graphs with $n$ vertices and $m$ edges; (4) planar graphs with $n$ vertices and $m$ edges; and (5) graphs with $n$ vertices and no $K_5$-minor or no $K_{3,3}$-minor. For example, the maximum number of cliques in a planar graph with $n$ vertices is $8(n-2)$.

Complexity results on graphs with few cliques

Graphs and Algorithms A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.

A map colour theorem for the union of graphs

In 1890 Heawood [Map colour theorem, Quart. J. Pure Appl. Math. 24 (1890) 332–338] established an upper bound for the chromatic number of a graph embedded on a surface of Euler genus g ⩾ 1 . This upper bound became known as the Heawood number H ( g ) . Almost a century later, Ringel [Map Color Theorem, Springer, New York, 1974] and Ringel and Youngs [Solution of the Heawood map-coloring problem, Proc. Nat. Acad. Sci. USA 60 (1968) 438–445] proved that the Heawood number H ( g ) is in fact the maximum chromatic number as well as the maximum clique number of graphs embedded on a surface of Euler genus g ⩾ 1 besides the Klein bottle. In this paper, we present a Heawood-type formula for the edge disjoint union of two graphs that are embedded on a given surface Σ . More precisely, we determine the number H 2 ( Σ ) such that if a graph G embedded on Σ is the edge disjoint union of two graphs G 1 and G 2 , then ω ( G 1 ) + ω ( G 2 ) ⩽ χ ( G 1 ) + χ ( G 2 ) ⩽ H 2 ( Σ ) . Similar to the results of Ringel and Ringel and Youngs, we show that this bound is sharp for all but at most one non-orientable surface Σ .

On the complexity of cell flipping in permutation diagrams and multiprocessor scheduling problems

Permutation diagrams have been used in circuit design to model a set of single point nets crossing a channel, where the minimum number of layers needed to realize the diagram equals the clique number ω ( G ) of its permutation graph, the value of which can be calculated in O ( n log n ) time. We consider a generalization of this model motivated by “standard cell” technology in which the numbers on each side of the channel are partitioned into consecutive subsequences, or cells, each of which can be left unchanged or flipped (i.e., reversed). We ask, for what choice of flippings will the resulting clique number be minimum or maximum. We show that when one side of the channel is fixed (no flipping), an optimal flipping for the other side can be found in O ( n log n ) time for the maximum clique number, and that when both sides are free this can be solved in O ( n 2 ) time. We also prove NP-completeness of finding a flipping that gives a minimum clique number, even when one side of the channel is fixed, and even when the size of the cells is restricted to be less than a small constant. Moreover, since the complement of a permutation graph is also a permutation graph, the same complexity results hold for the stable set (independence) number. In the process of the NP-completeness proof we also prove NP-completeness of a restricted variant of a scheduling problem. This new NP-completeness result may be of independent interest.