44,230 research outputs found
On Products and Line Graphs of Signed Graphs, their Eigenvalues and Energy
In this article we examine the adjacency and Laplacian matrices and their
eigenvalues and energies of the general product (non-complete extended -sum,
or NEPS) of signed graphs. We express the adjacency matrix of the product in
terms of the Kronecker matrix product and the eigenvalues and energy of the
product in terms of those of the factor signed graphs. For the Cartesian
product we characterize balance and compute expressions for the Laplacian
eigenvalues and Laplacian energy. We give exact results for those signed
planar, cylindrical and toroidal grids which are Cartesian products of signed
paths and cycles.
We also treat the eigenvalues and energy of the line graphs of signed graphs,
and the Laplacian eigenvalues and Laplacian energy in the regular case, with
application to the line graphs of signed grids that are Cartesian products and
to the line graphs of all-positive and all-negative complete graphs.Comment: 30 page
Complexity of planar signed graph homomorphisms to cycles
We study homomorphism problems of signed graphs. A signed graph is an
undirected graph where each edge is given a sign, positive or negative. An
important concept for signed graphs is the operation of switching at a vertex,
which is to change the sign of each incident edge. A homomorphism of a graph is
a vertex-mapping that preserves the adjacencies; in the case of signed graphs,
we also preserve the edge-signs. Special homomorphisms of signed graphs, called
s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the
mapping, to perform any number of switchings on the source signed graph. This
concept has been extensively studied, and a full complexity classification
(polynomial or NP-complete) for s-homomorphism to a fixed target signed graph
has recently been obtained. Such a dichotomy is not known when we restrict the
input graph to be planar (not even for non-signed graph homomorphisms).
We show that deciding whether a (non-signed) planar graph admits a
homomorphism to the square of a cycle with , or to the circular
clique with , are NP-complete problems. We use these
results to show that deciding whether a planar signed graph admits an
s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is
unbalanced if it has an odd number of negative edges). We deduce a complete
complexity dichotomy for the planar s-homomorphism problem with any signed
cycle as a target.
We also study further restrictions involving the maximum degree and the girth
of the input signed graph. We prove that planar s-homomorphism problems to
signed cycles remain NP-complete even for inputs of maximum degree~ (except
for the case of unbalanced -cycles, for which we show this for maximum
degree~). We also show that for a given integer , the problem for signed
bipartite planar inputs of girth is either trivial or NP-complete.Comment: 17 pages, 10 figure
Computing paths and cycles in biological interaction graphs
<p>Abstract</p> <p>Background</p> <p>Interaction graphs (signed directed graphs) provide an important qualitative modeling approach for Systems Biology. They enable the analysis of causal relationships in cellular networks and can even be useful for predicting qualitative aspects of systems dynamics. Fundamental issues in the analysis of interaction graphs are the enumeration of paths and cycles (feedback loops) and the calculation of shortest positive/negative paths. These computational problems have been discussed only to a minor extent in the context of Systems Biology and in particular the shortest signed paths problem requires algorithmic developments.</p> <p>Results</p> <p>We first review algorithms for the enumeration of paths and cycles and show that these algorithms are superior to a recently proposed enumeration approach based on elementary-modes computation. The main part of this work deals with the computation of shortest positive/negative paths, an NP-complete problem for which only very few algorithms are described in the literature. We propose extensions and several new algorithm variants for computing either exact results or approximations. Benchmarks with various concrete biological networks show that exact results can sometimes be obtained in networks with several hundred nodes. A class of even larger graphs can still be treated exactly by a new algorithm combining exhaustive and simple search strategies. For graphs, where the computation of exact solutions becomes time-consuming or infeasible, we devised an approximative algorithm with polynomial complexity. Strikingly, in realistic networks (where a comparison with exact results was possible) this algorithm delivered results that are very close or equal to the exact values. This phenomenon can probably be attributed to the particular topology of cellular signaling and regulatory networks which contain a relatively low number of negative feedback loops.</p> <p>Conclusion</p> <p>The calculation of shortest positive/negative paths and cycles in interaction graphs is an important method for network analysis in Systems Biology. This contribution draws the attention of the community to this important computational problem and provides a number of new algorithms, partially specifically tailored for biological interaction graphs. All algorithms have been implemented in the <it>CellNetAnalyzer </it>framework which can be downloaded for academic use at <url>http://www.mpi-magdeburg.mpg.de/projects/cna/cna.html</url>.</p
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