Recognizing hyperelliptic graphs in polynomial time

Recently, a new set of multigraph parameters was defined, called "gonalities". Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1 are trees. We consider so-called "hyperelliptic graphs" (multigraphs of gonality 2) and provide a safe and complete sets of reduction rules for such multigraphs, showing that for three of the flavors of gonality, we can recognize hyperelliptic graphs in O(n log n+m) time, where n is the number of vertices and m the number of edges of the multigraph.Comment: 33 pages, 8 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

Local topological data analysis to uncover the global structure of data approaching graph-structured topologies

Gene expression data of differentiating cells, galaxies distributed in space, and earthquake locations, all share a common property: they lie close to a graph-structured topology in their respective spaces [1, 4, 9, 10, 20], referred to as one-dimensional stratified spaces in mathematics. Often, the uncovering of such topologies offers great insight into these data sets. However, methods for dimensionality reduction are clearly inappropriate for this purpose, and also methods from the relatively new field of Topological Data Analysis (TDA) are inappropriate, due to noise sensitivity, computational complexity, or other limitations. In this paper we introduce a new method, termed Local TDA (LTDA), which resolves the issues of pre-existing methods by unveiling (global) graph-structured topologies in data by means of robust and computationally cheap local analyses. Our method rests on a simple graph-theoretic result that enables one to identify isolated, end-, edge- and multifurcation points in the topology underlying the data. It then uses this information to piece together a graph that is homeomorphic to the unknown one-dimensional stratified space underlying the point cloud data. We evaluate our method on a number of artificial and real-life data sets, demonstrating its superior effectiveness, robustness against noise, and scalability. Code related to this paper is available at: https://bitbucket.org/ghentdatascience/gltda-public

Network Decontamination

The Network Decontamination problem consists in coordinating a team of mobile agents in order to clean a contaminated network. The problem is actually equivalent to tracking and capturing an invisible and arbitrarily fast fugitive. This problem has natural applications in network security in computer science or in robotics for search or pursuit-evasion missions. In this Chapter, we focus on networks modeled by graphs. Many different objectives have been studied in this context, the main one being the minimization of the number of mobile agents necessary to clean a contaminated network. Another important aspect is that this optimization problem has a deep graph-theoretical interpretation. Network decontamination and, more precisely, graph searching models, provide nice algorithmic interpretations of fundamental concepts in the Graph Minors theory by Robertson and Seymour. For all these reasons, graph searching variants have been widely studied since their introduction by Breish (1967) and mathematical formaliza-tions by Parsons (1978) and Petrov (1982). This chapter consists of an overview of algorithmic results on graph de-contamination and graph searching