Showing papers on "K-tree published in 2008"

Large maximal cliques enumeration in sparse graphs

Abstract: Here we study a variant of maximal clique enumeration problem by incorporating a minimum size criterion. We describe preprocessing techniques to reduce the graph size. This is of practical interest since enumerating maximal cliques is a computationally hard problem and the execution time increases rapidly with the input size. We discuss basics of an algorithm for enumerating large maximal cliques which exploits the constraint on minimum size of the desired maximal cliques. Social networks are prime examples of large sparse graphs where enumerating large maximal cliques is of interest. We present experimental results on the social network formed by the call detail records of one of the world's largest telecom service providers. Our results show that the preprocessing methods achieve significant reduction in the graph size. We also characterize the execution behaviour of our large maximal clique enumeration algorithm.

Method for enumerating cliques

Abstract: Techniques for enumerating at least one maximal clique are provided. The techniques include obtaining data, wherein the data comprises a graph, obtaining a user-specified minimum size restriction on at least one maximal clique of interest, filtering the data using the user-specified minimum size restriction to reduce graph size, and enumerating at least one maximal clique from the graph provided that at least one maximal clique exists above the user-specified minimum size restriction.

A Characterization of the degree sequences of 2-trees

Abstract: A graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbors are adjacent, and G- v is a 2- tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for recognizing and realizing degree sequences of 2-trees. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:191-209, 2008 This research was initiated at The 21st Bellairs Winter Workshop on Computational Geometry, January 27–February 3, 2006.

Enumerating Isolated Cliques in Synthetic and Financial Networks

Abstract: We do computational studies concerning the enumeration of maximal isolated cliques in graphs Isolation, as recently introduced, measures the degree of connectedness of the cliques to the rest of the graph Isolation helps both in getting faster algorithms than for the enumeration of maximal general cliques and in filtering out cliques with special semantics We perform experiments with synthetic graphs (in the G n,m,p model) and financial networks, proposing the enumeration of isolated cliques as a useful instrument in analyzing financial networks

Mining of Frequent Externally Extensible Outerplanar Graph Patterns

Abstract: An outerplanar graph is a planar graph which can be embedded in the plane in such a way that all of vertices lie on the outer boundary. Many chemical compounds are known to be expressed by outerplanar graphs. In this paper, firstly, we introduce an externally extensible outerplanar graph pattern (eeo-graph pattern for short) as a graph pattern common to a finite set of outerplanar graphs like a dataset of chemical compounds. The eeo-graph pattern can express a substructure common to blocks which appear in outerplanar graph structured data. Secondly, we propose a data mining algorithm for enumerating all maximal frequent eeo-graph patterns from a finite set of outerplanar graphs. Finally, we report experimental results on a chemical dataset.

Enumeration of Perfect Sequences of Chordal Graph

Abstract: A graph is chordal if and only if it has no chordless cycle of length more than three. The set of maximal cliques in a chordal graph admits special tree structures called clique trees. A perfect sequence is a sequence of maximal cliques obtained by using the reverse order of repeatedly removing the leaves of a clique tree. This paper addresses the problem of enumerating all the perfect sequences. Although this problem has statistical applications, no efficient algorithm has been proposed. There are two difficulties with developing this type of algorithms. First, a chordal graph does not generally have a unique clique tree. Second, a perfect sequence can normally be generated by two or more distinct clique trees. Thus it is hard using a straightforward way to generate perfect sequences from each possible clique tree. In this paper, we propose a method to enumerate perfect sequences without constructing clique trees. As a result, we have developed the first polynomial delay algorithm for dealing with this problem. In particular, the time complexity of the algorithm on average is O(1) for each perfect sequence.

Learning inclusion-optimal chordal graphs

Abstract: Chordal graphs can be used to encode dependency models that are representable by both directed acyclic and undirected graphs. This paper discusses a very simple and efficient algorithm to learn the chordal structure of a probabilistic model from data. The algorithm is a greedy hill-climbing search algorithm that uses the inclusion boundary neighborhood over chordal graphs. In the limit of a large sample size and under appropriate hypotheses on the scoring criterion, we prove that the algorithm will find a structure that is inclusion-optimal when the dependency model of the data-generating distribution can be represented exactly by an undirected graph. The algorithm is evaluated on simulated datasets.

A clique-difference encoding scheme for labelled k-path graphs

Abstract: We present in this paper a codeword for labelled k-path graphs. Structural properties of this codeword are investigated, leading to the solution of two important problems: determining the exact number of labelled k-path graphs with n vertices and locating a hamiltonian path in a given k-path graph in time O(n). The corresponding encoding scheme is also presented, providing linear-time algorithms for encoding and decoding.

Binary positive semidefinite matrices and associated integer polytopes

Abstract: We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytopes. Several important and well-known integer polytopes -- the cut, boolean quadric, multicut and clique partitioning polytopes -- are shown to arise as projections of binary psd polytopes. Finally, we present various valid inequalities for binary psd polytopes, and show how they relate to inequalities known for the simpler polytopes mentioned. Along the way, we answer an open question in the literature on the maxcut problem, by showing that the so-called k-gonal inequalities define a polytope.

Solving the Maximum Clique Problem on a Class of Network Graphs, With Application to Social Networks

Abstract: : Social network analysis frequently uses the idea of a clique in a network to identify key subgroups of highly-connected members of the network. We formulate the maximum clique problem on undirected graphs and develop two algorithms to solve it: a pruning algorithm and an enumeration algorithm. The pruning algorithm successively improves an upper bound on the clique number of a graph, and the enumeration algorithm successively finds larger and larger cliques in the graph. Both terminate with a maximum clique in the graph, and, when run together, provide an interval of uncertainty on the size of a maximum clique in a graph that converges to zero. We apply our algorithms to real examples in the modeling of terrorist social networks, and determine that our algorithms are efficient and practical for problems of moderate size.

Algorithmic Generation of Graphs of Branch-width ≤ k

Abstract: Branchwidth is a connectivity parameter of graphs closely related to treewidth. Graphs of treewidth at most k can be generated algorithmically as the subgraphs of k-trees. In this paper, we investigate the family of edge-maximal graphs of branchwidth k, that we call k-branches. The k-branches are, just as the k-trees, a subclass of the chordal graphs where all minimal separators have size k. However, a striking difference arises when considering subgraph-minimal members of the family. Whereas K_{k+1} is the only subgraph-minimal k-tree, we show that for any k ≤ 7 a minimal k-branch having q maximal cliques exists for any value of q different than 3 and 5, except for k=8,q=2. We characterize subgraph-minimal k-branches for all values of k. Our investigation leads to a generation algorithm, that adds one or two new maximal cliques in each step, producing exactly the k-branches.

On Greedy Clique Decompositions and Set Representations of Graphs

Abstract: In 1994 S. McGuinness showed that any greedy clique decomposition of an n-vertex graph has at most ⌊n 2 /4⌋ cliques (The greedy clique decomposition of a graph, J. Graph Theory 18 (1994) 427-430), where a clique decomposition means a clique partition of the edge set and a greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. This result solved a conjecture by P. Winkler. A multifamily set representation of a simple graph G is a family of sets, not necessarily distinct, each member of which represents a vertex in G, and the intersection of two sets is non-empty if and only if two corresponding vertices in G are adjacent. It is well known that for a graph G, there is a one-to-one correspondence between multifamily set representations and clique coverings of the edge set. Further for a graph one may have a one-to-one correspondence between particular multifamily set representations with intersection size at most one and clique partitions of the edge set. In this paper, we study for an n-vertex graph the variant of the set representations using a family of distinct sets, including the greedy way to get the corresponding clique partition of the edge set of the graph. Similarly, in this case, we obtain a result that any greedy clique decomposition of an n-vertex graph has at most ⌊n 2 /4⌋ cliques.

Study of Chordal graphs

Abstract: THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF Bottor of $t)t(ogapI)p IN APPLIED MATHEMATICS