Showing papers on "K-tree published in 2010"
TL;DR: A benchmark network is shown where clique graphs find the overlapping communities accurately while vertex partition methods fail, and how a clique graph may be exploited.
Abstract: It is shown how to construct a clique graph in which properties of cliques of a fixed order in a given graph are represented by vertices in a weighted graph. Various definitions and motivations for these weights are given. The detection of communities or clusters is used to illustrate how a clique graph may be exploited. In particular a benchmark network is shown where clique graphs find the overlapping communities accurately while vertex partition methods fail.
160 citations
01 Mar 2010
TL;DR: An optimized branch-and-bound algorithm is developed to find top-k maximal cliques in an uncertain graph, which adopts efficient pruning rules, a new searching strategy and effective preprocessing methods.
Abstract: Existing studies on graph mining focus on exact graphs that are precise and complete. However, graph data tends to be uncertain in practice due to noise, incompleteness and inaccuracy. This paper investigates the problem of finding top-k maximal cliques in an uncertain graph. A new model of uncertain graphs is presented, and an intuitive measure is introduced to evaluate the significance of vertex sets. An optimized branch-and-bound algorithm is developed to find top-k maximal cliques, which adopts efficient pruning rules, a new searching strategy and effective preprocessing methods. The extensive experimental results show that the proposed algorithm is very efficient on real uncertain graphs, and the top-k maximal cliques are very useful for real applications, e.g. protein complex prediction.
115 citations
04 Mar 2010
TL;DR: In this article, it was shown that given an n-vertex graph G together with its set of potential maximal cliques, and an integer t, it is possible to find a maximum induced subgraph of treewidth t in G in time O(n −1.734601^n)
Abstract: Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulation problems including Minimum Fill-in and Treewidth. We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal cliques, and an integer t, it is possible in time the number of potential maximal cliques times $O(n^{O(t)})$ to find a maximum induced subgraph of treewidth t in G and for a given graph F of treewidth t, to decide if G contains an induced subgraph isomorphic to F.
Combined with an improved algorithm enumerating all potential maximal cliques in time $O(1.734601^n)$, this yields that both the problems are solvable in time $1.734601^n$ * $n^{O(t)}$.
108 citations
01 Jan 2010
TL;DR: This paper addresses the problem of enumerating all pseudo cliques for a given graph and shows that it seems to be difficult to obtain polynomial time algorithms using straightforward divide and conquer approaches, and proposes an algorithm based on reverse search.
Abstract: The problem of finding dense structures in a given graph is quite basic in informatics including data mining and data engineering. Clique is a popular model to represent dense structures, and widely used because of its simplicity and ease in handling. Pseudo cliques are natural extension of cliques which are subgraphs obtained by removing small number of edges from cliques. We here define a pseudo clique by a subgraph such that the ratio of the number of its edges compared to that of the clique with the same number of vertices is no less than a given threshold value. In this paper, we address the problem of enumerating all pseudo cliques for a given graph and a threshold value. We first show that it seems to be difficult to obtain polynomial time algorithms using straightforward divide and conquer approaches. Then, we propose a polynomial time, polynomial delay in precise, algorithm based on reverse search. The time complexity for each pseudo clique is O(Δlog |V|+min {Δ 2,|V|+|E|}). Computational experiments show the efficiency of our algorithm for both randomly generated graphs and practical graphs.
79 citations
13 Dec 2010
TL;DR: dMaximalCliques is a distributed algorithm which can obtain clique information from million-node graphs within a few minutes on an 80-node computer cluster and the distribution of the size of maximal cliques in a graph is proposed as a new measure for measuring the structural properties of a graph.
Abstract: Clique detection and analysis is one of the fundamental problems in graph theory. However, as the size of graphs increases (e.g., those of social networks), it becomes difficult to conduct such analysis using existing sequential algorithms due to the computation and memory limitation. In this paper, we present a distributed algorithm, dMaximalCliques, which can obtain clique information from million-node graphs within a few minutes on an 80-node computer cluster. dMaximalCliques is a distributed algorithm for share-nothing systems, such as racks of clusters. We use very large scale real and synthetic graphs in the experimental studies to prove the efficiency of the algorithm. In addition, we propose to use the distribution of the size of maximal cliques in a graph (Maximal Clique Distribution) as a new measure for measuring the structural properties of a graph and for distinguishing different types of graphs. Meanwhile, we find that this distribution can be well fitted by lognormal distribution.
28 citations
TL;DR: This paper considers the problem of clustering a set of items into subsets whose sizes are bounded from above and below and proposes an integer programming model for solving it and analyzes the structure of the corresponding polytope to prove several results concerning the facial structure.
26 citations
TL;DR: It is proved that the normality of cut polytopes of graphs is a minor closed property, and by using this result, the authors have large classes of normal cutpolytopes.
22 citations
TL;DR: This paper characterize all instances where the bounds are achieved, and determine exactly the independence polynomials of several classes of k-tree related graphs, and generalize several related results known before.
21 citations
TL;DR: This paper proposes a simple random model for generating scale free k-trees, and experimental results indicate that the resultant k-Trees have extremely small diameter, proportional to o(log n), where n is the number of vertices in the k-tree, and the o(1) term is a function of k.
Abstract: Scale free graphs have attracted attention as their non-uniform structure that can be used as a model for many social networks including the WWW and the Internet In this paper, we propose a simple random model for generating scale free k-trees For any fixed integer k, a k-tree consists of a generalized tree parameterized by k, and is one of the basic notions in the area of graph minors Our model is quite simple and natural; it first picks a maximal clique of size k + 1 uniformly at random, it then picks k vertices in the clique uniformly at random, and adds a new vertex incident to the k vertices That is, the model only makes uniform random choices twice per vertex Then (asymptotically) the distribution of vertex degree in the resultant k-tree follows a power law with exponent 2 + 1/k, the k-tree has a large clustering coefficient, and the diameter is small Moreover, our experimental results indicate that the resultant k-trees have extremely small diameter, proportional to o(log n), where n is the number of vertices in the k-tree, and the o(1) term is a function of k
19 citations
TL;DR: A new one-phase algorithm based on properties of the lexicographic breadth-first search is proposed and a characterization for planar chordal graphs is presented; using the proposed algorithm as an initial step, the implementation of the recognition algorithm becomes trivial.
18 citations
TL;DR: The first polynomial delay algorithm for dealing with the problem of enumerating all the perfect sequences is proposed, and the time complexity of the algorithm on average is O(1) for each perfect sequence.
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12 Oct 2010
TL;DR: The main focus of as mentioned in this paper is to evaluate kr(n, δ), the minimal number of r-cliques in graphs with n vertices and minimum degree δ.
Abstract: The main focus of this thesis is to evaluate kr(n, δ), the minimal number of r-cliques in graphs with n vertices and minimum degree δ. A fundamental result in Graph Theory states that a triangle-free graph of order n has at most n/4 edges. Hence, a triangle-free graph has minimum degree at most n/2, so if k3(n, δ) = 0 then δ ≤ n/2. For n/2 ≤ δ ≤ 4n/5, I have evaluated kr(n, δ) and determined the structures of the extremal graphs. For δ ≥ 4n/5, I give a conjecture on kr(n, δ), as well as the structures of these extremal graphs. Moreover, I have proved various partial results that support this conjecture. Let k r (n, δ) be the analogous version of kr(n, δ) for regular graphs. Notice that there exist n and δ such that kr(n, δ) = 0 but k reg r (n, δ) > 0. For example, a theorem of Andrasfai, Erdős and Sos states that any triangle-free graph of order n with minimum degree greater than 2n/5 must be bipartite. Hence k3(n, bn/2c) = 0 but k 3 (n, bn/2c) > 0 for n odd. I have evaluated the exact value k 3 (n, δ) for δ between 2n/5 + 12 √ n/5 and n/2 and determined the structure of these extremal graphs. At the end of the thesis, I investigate a question in Ramsey Theory. The Ramsey number Rk(G) of a graph G is the minimum number N , such that any edge colouring of KN with k colours contains a monochromatic copy of G. The constrained Ramsey number f(G, T ) of two graphs G and T is the minimum number N such that any edge colouring of KN with any number of colours contains a monochromatic copy of G or a rainbow copy of T . It turns out that these two quantities are closely related when T is a matching. Namely, for almost all graphs G, f(G, tK2) = Rt−1(G) for t ≥ 2.
TL;DR: By enumerating only the difference set between the baseline and perturbed graphs' MCEs, the computational cost of enumerating the maximal cliques of the perturbed graph can be reduced.
TL;DR: In this paper, Deng et al. proved Chvatal's conjecture on maximal stable sets and maximal cliques in graphs and showed that this conjecture generalizes to planar graphs.
Abstract: X. Deng et al. proved Chvatal’s conjecture on maximal stable sets and maximal cliques in graphs. G. Ding made a conjecture to generalize Chvatal’s conjecture. The purpose of this paper is to prove this conjecture in planar graphs and the complement of planar graphs.
TL;DR: This paper gives sufficient conditions for a graph to have a k-tree containing specified vertices and shows some new results on a spanning k- tree as corollaries of the above theorem.
Abstract: A k-tree is a tree with maximum degree at most k. In this paper, we give sufficient conditions for a graph to have a k-tree containing specified vertices. Let k be an integer with k > 3. Let G be a graph of order n and let \({S \subseteq V(G)}\) with κ(S) ≥ 1. Suppose that for every l > κ(S), there exists an integer t such that \({1 \le t \leq (k-1)l+2 - \lfloor \frac{l-1}{k} \rfloor}\) and the degree sum of any t independent vertices of S is at least n + tl − kl − 1. Then G has a k-tree containing S. We also show some new results on a spanning k-tree as corollaries of the above theorem.
TL;DR: The branch number of G is introduced, a measure of how complex the k-tree is, and it is seen by the definition that the branch number is easier to calculate and to work with than the toughness of a graph.
01 Jan 2010
TL;DR: In this paper, the clique covering number of a graph G is defined as the set of cliques of G in which each edge of G is contained in at least one clique.
Abstract: A clique covering of a graph G is a set of cliques of G in which each edge of G is contained in at least one clique. The smallest cardinality of clique coverings of G is called the clique covering number of G. A glued graph results from combining two nontrivial vertex-disjoint graphs by identifying nontrivial connected isomorphic subgraphs of both graphs. Such subgraphs are referred to as the clones. The two nontrivial vertexdisjoint graphs are referred to the original graphs. In this paper, we investigate bounds of clique covering numbers of glued graphs at clone which is isomorphic to Kn in terms of clique covering numbers of their original graphs, and give a characterization of a glued graph with the clique covering number of each possible value.
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TL;DR: A structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of edge-disjoint spanning trees in G are provided.
Abstract: We provide a structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of edge-disjoint spanning trees in G. These graphs are of interest for the k-tree protocol of Itai and Rodeh [Inform. and Comput. 79 (1988), 43-59].
TL;DR: A tight lower bound is established on the signed clique-transversal number for a regular graph with clique number at most 4 and the decision problem corresponding to the problem of computing is NP-complete even when restricted to doubly chordal graphs.
Abstract: A function f: V→{-1,+1}, defined on the vertices of a graph G, is a signed clique-transversal function (SCTF) if ∑u∈V(C)f(u)≥1 for every clique C of G. The weight of an SCTF is w(f)=∑v∈V(G)f(v). The signed clique-transversal number, denoted [image omitted] , is the minimum weight of an SCTF of G. The signed clique-transversal problem is to find an SCTF of minimum weight for G. In this paper, we establish a tight lower bound on the signed clique-transversal number for a regular graph with clique number at most 4. Furthermore, we show that the decision problem corresponding to the problem of computing [image omitted] is NP-complete even when restricted to doubly chordal graphs. Also, we prove that the signed clique-transversal problem can be solved in linear time for a strongly chordal graph if its strong elimination ordering is given.
19 May 2010
TL;DR: Three methods that could be used to estimate the number of cliques in a random graph are examined, one method is based on sampling, another on probability arguments and the third uses curve fitting.
Abstract: This paper examines methods for predicting and estimating the number of maximal cliques in a random graph. A clique is a subgraph where each vertex is connected to every other vertex in the subgraph. A maximal clique is a clique which is not a proper subgraph of another clique. There are many algorithms that enumerate all maximal cliques in a graph, but since the task can take exponential time, there are practical limits on the size of the input. In this paper, we examine three methods that could be used to estimate the number of cliques in a random graph. One method is based on sampling, another on probability arguments and the third uses curve fitting. We compare the methods for accuracy and efficiency.
01 Jan 2010
TL;DR: In this article, the authors studied the class of self-complementary (sc) weakly chordal graphs and derived O(m3) time algorithms to determine whether a graph is weakly chordal or not.
Abstract: The class of self-complementary(sc) weakly chordal graphs is studied,which is a generalization of self-complementary chordal graphs,lower and upper bounds for the number of two-pairs in sc weakly chordal graphs have been obtained.The recognition problem on the self-complementary weakly chordal graphs is discussed.In particular,an O(m3) time algorithm which determines whether a self-complementary graph is weakly chordal or not is discussed.Moreover,self-complementary weakly chordal graphs with at most 17 vertices is in catalogue.
TL;DR: A sufficient condition using the condition on forbidden subgraphs for a graph G to have a spanning k-tree is given.
TL;DR: The maximal cliques generation algorithm is based on generating all maximal paths in a directed acyclic graph, and an algorithm for this problem is presented that uses O log (p) communication rounds with O (m /p ) local computation for each maximal path.
Abstract: We present parallel algorithms on the BSP/CGM model, with p processors, to count and generate all the maximal cliques of a circle graph with n vertices and m edges. To count the number of all the maximal cliques, without actually generating them, our algorithm requires O (log p ) communication rounds with O (nm /p ) local computation time. We also present an algorithm to generate the first maximal clique in O (log p ) communication rounds with O (nm /p ) local computation, and to generate each one of the subsequent maximal cliques this algorithm requires O (log p ) communication rounds with O (m /p ) local computation. The maximal cliques generation algorithm is based on generating all maximal paths in a directed acyclic graph, and we present an algorithm for this problem that uses O log (p) communication rounds with O (m /p ) local computation for each maximal path. We also show that the presented algorithms can be extended to the CREW PRAM model.