616 research outputs found

    On the Recognition of Fuzzy Circular Interval Graphs

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    Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of claw-free graphs. In this paper, we provide a polynomial-time algorithm for recognizing such graphs, and more importantly for building a suitable representation.Comment: 12 pages, 2 figure

    Polynomial-time algorithm for Maximum Weight Independent Set on P6P_6-free graphs

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    In the classic Maximum Weight Independent Set problem we are given a graph GG with a nonnegative weight function on vertices, and the goal is to find an independent set in GG of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any P6P_6-free graph, that is, a graph that has no path on 66 vertices as an induced subgraph. This improves the polynomial-time algorithm on P5P_5-free graphs of Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on P6P_6-free graphs of Lokshtanov et al (SODA 2016). The main technical contribution leading to our main result is enumeration of a polynomial-size family F\mathcal{F} of vertex subsets with the following property: for every maximal independent set II in the graph, F\mathcal{F} contains all maximal cliques of some minimal chordal completion of GG that does not add any edge incident to a vertex of II

    Modularity measure of networks with overlapping communities

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    In this paper we introduce a non-fuzzy measure which has been designed to rank the partitions of a network's nodes into overlapping communities. Such a measure can be useful for both quantifying clusters detected by various methods and during finding the overlapping community-structure by optimization methods. The theoretical problem referring to the separation of overlapping modules is discussed, and an example for possible applications is given as well

    Edge Clique Cover of Claw-free Graphs

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    The smallest number of cliques, covering all edges of a graph G G , is called the (edge) clique cover number of G G and is denoted by cc(G) cc(G) . It is an easy observation that for every line graph G G with n n vertices, cc(G)ncc(G)\leq n . G. Chen et al. [Discrete Math. 219 (2000), no. 1--3, 17--26; MR1761707] extended this observation to all quasi-line graphs and questioned if the same assertion holds for all claw-free graphs. In this paper, using the celebrated structure theorem of claw-free graphs due to Chudnovsky and Seymour, we give an affirmative answer to this question for all claw-free graphs with independence number at least three. In particular, we prove that if G G is a connected claw-free graph on n n vertices with α(G)3 \alpha(G)\geq 3 , then cc(G)n cc(G)\leq n and equality holds if and only if G G is either the graph of icosahedron, or the complement of a graph on 1010 vertices called twister or the pthp^{th} power of the cycle Cn C_n , for 1p(n1)/31\leq p \leq \lfloor (n-1)/3\rfloor .Comment: 74 pages, 4 figure

    Searching for network modules

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    When analyzing complex networks a key target is to uncover their modular structure, which means searching for a family of modules, namely node subsets spanning each a subnetwork more densely connected than the average. This work proposes a novel type of objective function for graph clustering, in the form of a multilinear polynomial whose coefficients are determined by network topology. It may be thought of as a potential function, to be maximized, taking its values on fuzzy clusterings or families of fuzzy subsets of nodes over which every node distributes a unit membership. When suitably parametrized, this potential is shown to attain its maximum when every node concentrates its all unit membership on some module. The output thus is a partition, while the original discrete optimization problem is turned into a continuous version allowing to conceive alternative search strategies. The instance of the problem being a pseudo-Boolean function assigning real-valued cluster scores to node subsets, modularity maximization is employed to exemplify a so-called quadratic form, in that the scores of singletons and pairs also fully determine the scores of larger clusters, while the resulting multilinear polynomial potential function has degree 2. After considering further quadratic instances, different from modularity and obtained by interpreting network topology in alternative manners, a greedy local-search strategy for the continuous framework is analytically compared with an existing greedy agglomerative procedure for the discrete case. Overlapping is finally discussed in terms of multiple runs, i.e. several local searches with different initializations.Comment: 10 page
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