Constructing level-2 phylogenetic networks from triplets

Jansson and Sung showed that, given a dense set of input triplets T (representing hypotheses about the local evolutionary relationships of triplets of species), it is possible to determine in polynomial time whether there exists a level-1 network consistent with T, and if so to construct such a network. They also showed that, unlike in the case of trees (i.e. level-0 networks), the problem becomes NP-hard when the input is non-dense. Here we further extend this work by showing that, when the set of input triplets is dense, the problem is even polynomial-time solvable for the construction of level-2 networks. This shows that, assuming density, it is tractable to construct plausible evolutionary histories from input triplets even when such histories are heavily non-tree like. This further strengthens the case for the use of triplet-based methods in the construction of phylogenetic networks. We also show that, in the non-dense case, the level-2 problem remains NP-hard

A Vizing-type theorem for matching forests

A well-known Theorem of Vizing states that one can colour the edges of a graph by Δ + α colours, such that edges of the same colour form a matching. Here, Δ denotes the maximum degree of a vertex, and α the maximum multiplicity of an edge in the graph. An analogue of this Theorem for directed graphs was proved by Frank. It states that one can colour the arcs of a digraph by Δ + α colours, such that arcs of the same colour form a branching. For a digraph, A denotes the maximum indegree of a vertex, and a the maximum multiplicity of an arc. We prove a common generalization of the above two theorems concerning the colouring of mixed graphs (these are graphs having both directed and undirected edges) in such a way that edges of the same colour form a matching forest

Conditions for β-perfectness

A β-perfect graph is a simple graph G such that χ(G') = β(G') for every induced subgraph G' of G, where χ(G') is the chromatic number of G', and β(G') is defined as the maximum over all induced subgraphs H of G' of the minimum vertex degree in H plus 1 (i.e., δ(H)+1). The vertices of a β-perfect graph G can be coloured with χ(G) colours in polynomial time (greedily). The main purpose of this paper is to give necessary and sufficient conditions, in terms of forbidden induced subgraphs, for a graph to be β-perfect. We give new sufficient conditions and make improvements to sufficient conditions previously given by others. We also mention a necessary condition which generalizes the fact that no β-perfect graph contains an even hole

Reconstructing a phylogenetic level-1 network from quartets

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on n taxa from the set of all quartets containing a certain fixed taxon, in O(n^3) time. We also present a more general method which can handle more diverse quartet data, but which takes O(n^6) time. Both methods proceed by solving a certain system of linear equations over the two-element field GF(2) . For a general dense quartet set, i.e. a set containing at least one quartet on every four taxa, our O(n^6) algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an O(n^2) -sized certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set, and more particularly regarding the complexity of constructing a cyclic ordering of taxa consistent with a dense quartet set

Virtual private network design : a proof of the tree routing conjecture on ring networks

A basic question in virtual private network (VPN) design is if the symmetric version of the problem always has an optimal solution which is a tree network. An affirmative answer would imply that the symmetric VPN problem is solvable in polynomial time. We give an affirmative answer in case the communication network, within which the VPN must be created, is a circuit. This seems to be an important step towards answering the general question. The proof relies on a dual pair of linear programs and actually implies an even stronger property of VPNs. We show that this property also holds for some other special cases of the problem, in particular when the network is a tree of rings

Prefix reversals on binary and ternary strings

Given a permutation π, the application of prefix reversal f (i) to π reverses the order of the first i elements of π. The problem of Sorting By Prefix Reversals (also known as pancake flipping), made famous by Gates and Papadimitriou (Bounds for sorting by prefix reversal, Discrete Mathematics 27, pp. 47-57), asks for the minimum number of prefix reversals required to sort the elements of a given permutation. In this paper we study a variant of this problem where the prefix reversals act not on permutations but on strings over a fixed size alphabet. We determine the minimum number of prefix reversals required to sort binary and ternary strings, with polynomial-time algorithms for these sorting problems as a result; demonstrate that computing the minimum prefix reversal distance between two binary strings is NP-hard; give an exact expression for the prefix reversal diameter of binary strings, and give bounds on the prefix reversal diameter of ternary strings. We also consider a weaker form of sorting called grouping (of identical symbols) and give polynomial-time algorithms for optimally grouping binary and ternary strings. A number of intriguing open problems are also discussed

Inter-individual variability in habituation of anxiety-related responses within three mouse inbred strains

Inter-individual variability in behavioral and physiological response has become a well-established phenomenon in animal models of anxiety and other disorders. Such variability is even demonstrated within mouse inbred strains. A recent study showed that adaptive and non-adaptive anxiety phenotypes (measured as habituation and/or sensitization of anxiety responses) may differ within cohorts of 129 mice. This variability was expressed across both anxiety- and activity-related behavioral dimensions. These findings were based however on re-analysis of previously published data. The present study therefore aimed to empirically validate these findings in 129 mice. In addition, we assessed such inter-individuality in two other strains: BALB/c and C57BL/6. Males of three mouse inbred strains (BALB/c, C57BL/6 and 129S2) were behaviorally characterized through repeated exposure to a mild aversive stimulus (modified Hole Board, 4 consecutive trials). Behavioral observations were supplemented with assessment of circulating corticosterone levels. Clustering the individual response trajectories of behavioral and endocrine responses yielded two multidimensional response types of different adaptive value. Interestingly, these response types were displayed by individuals of all three strains. The response types differed significantly on anxiety and activity related behavioral dimensions but not on corticosterone concentrations. This study empirically confirms that adaptive capacities may differ within 129 cohorts. In addition, it extends this inter-individual variability in behavioral profiles to BALB/c and C57BL/6. Whether these two sub-types constitute differential anxiety phenotypes may differ per strain and requires further study