Orienting Graphs to Optimize Reachability

The paper focuses on two problems: (i) how to orient the edges of an undirected graph in order to maximize the number of ordered vertex pairs (x,y) such that there is a directed path from x to y, and (ii) how to orient the edges so as to minimize the number of such pairs. The paper describes a quadratic-time algorithm for the first problem, and a proof that the second problem is NP-hard to approximate within some constant 1+epsilon > 1. The latter proof also shows that the second problem is equivalent to ``comparability graph completion''; neither problem was previously known to be NP-hard

End vertices in containment interval graphs

An interval containment model of a graph maps vertices into intervals of a line in such a way that two vertices are adjacent if and only if the corresponding intervals are comparable under the inclusion relation. Graphs admitting an interval containment model are called containment interval graphs or CI graphs for short. A vertex v of a CI graph G is an end-vertex if there is an interval containment model of G in which the left endpoint of the interval corresponding to v is less than all other endpoints. In this work,we present a characterization of end-vertices in terms of forbidden induced subgraphs.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

End vertices in containment interval graphs

An interval containment model of a graph maps vertices into intervals of a line in such a way that two vertices are adjacent if and only if the corresponding intervals are comparable under the inclusion relation. Graphs admitting an interval containment model are called containment interval graphs or CI graphs for short. A vertex v of a CI graph G is an end-vertex if there is an interval containment model of G in which the left endpoint of the interval corresponding to v is less than all other endpoints. In this work,we present a characterization of end-vertices in terms of forbidden induced subgraphs.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

Differential analysis of biological networks

In cancer research, the comparison of gene expression or DNA methylation networks inferred from healthy controls and patients can lead to the discovery of biological pathways associated to the disease. As a cancer progresses, its signalling and control networks are subject to some degree of localised re-wiring. Being able to detect disrupted interaction patterns induced by the presence or progression of the disease can lead to the discovery of novel molecular diagnostic and prognostic signatures. Currently there is a lack of scalable statistical procedures for two-network comparisons aimed at detecting localised topological differences. We propose the dGHD algorithm, a methodology for detecting differential interaction patterns in two-network comparisons. The algorithm relies on a statistic, the Generalised Hamming Distance (GHD), for assessing the degree of topological difference between networks and evaluating its statistical significance. dGHD builds on a non-parametric permutation testing framework but achieves computationally efficiency through an asymptotic normal approximation. We show that the GHD is able to detect more subtle topological differences compared to a standard Hamming distance between networks. This results in the dGHD algorithm achieving high performance in simulation studies as measured by sensitivity and specificity. An application to the problem of detecting differential DNA co-methylation subnetworks associated to ovarian cancer demonstrates the potential benefits of the proposed methodology for discovering network-derived biomarkers associated with a trait of interest