616 research outputs found
On the Recognition of Fuzzy Circular Interval Graphs
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 -free graphs
In the classic Maximum Weight Independent Set problem we are given a graph
with a nonnegative weight function on vertices, and the goal is to find an
independent set in of maximum possible weight. While the problem is NP-hard
in general, we give a polynomial-time algorithm working on any -free
graph, that is, a graph that has no path on vertices as an induced
subgraph. This improves the polynomial-time algorithm on -free graphs of
Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on
-free graphs of Lokshtanov et al (SODA 2016). The main technical
contribution leading to our main result is enumeration of a polynomial-size
family of vertex subsets with the following property: for every
maximal independent set in the graph, contains all maximal
cliques of some minimal chordal completion of that does not add any edge
incident to a vertex of
Modularity measure of networks with overlapping communities
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
The smallest number of cliques, covering all edges of a graph , is
called the (edge) clique cover number of and is denoted by . It
is an easy observation that for every line graph with vertices,
. 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 is a
connected claw-free graph on vertices with , then and equality holds if and only if is either the graph of
icosahedron, or the complement of a graph on vertices called twister or
the power of the cycle , for .Comment: 74 pages, 4 figure
Searching for network modules
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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