The problem of constructing an optimal rooted phylogenetic network from an arbitrary set of rooted triplets is an NP-hard problem. In this paper, we present a heuristic algorithm called TripNet, which tries to construct a rooted phylogenetic network with the minimum number of reticulation nodes from an arbitrary set of rooted triplets. Despite of current methods that work for dense set of rooted triplets, a key innovation is the applicability of TripNet to non-dense set of rooted triplets. We prove some theorems to clarify the performance of the algorithm. To demonstrate the efficiency of TripNet, we compared TripNet with SIMPLISTIC. It is the only available software which has the ability to return some rooted phylogenetic network consistent with a given dense set of rooted triplets. But the results show that for complex networks with high levels, the SIMPLISTIC running time increased abruptly. However in all cases TripNet outputs an appropriate rooted phylogenetic network in an acceptable time. Also we tetsed TripNet on the Yeast data. The results show that Both TripNet and optimal networks have the same clustering and TripNet produced a level-3 network which contains only one more reticulation node than the optimal network.

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BackgroundUnderstanding the mechanisms by which hundreds of diverse cell types develop from a single mammalian zygote has been a central challenge of developmental biology. Conrad H. Waddington, in his metaphoric “epigenetic landscape” visualized the early embryogenesis as a hierarchy of lineage bifurcations. In each bifurcation, a single progenitor cell type produces two different cell lineages. The tristable dynamical systems are used to model the lineage bifurcations. It is also shown that a genetic circuit consisting of two auto-activating transcription factors (TFs) with cross inhibitions can form a tristable dynamical system.ResultsWe used gene expression profiles of pre-implantation mouse embryos at the single cell resolution to visualize the Waddington landscape of the early embryogenesis. For each lineage bifurcation we identified two clusters of TFs – rather than two single TFs as previously proposed – that had opposite expression patterns between the pair of bifurcated cell types. The regulatory circuitry among each pair of TF clusters resembled a genetic circuit of a pair of single TFs; it consisted of positive feedbacks among the TFs of the same cluster, and negative interactions among the members of the opposite clusters. Our analyses indicated that the tristable dynamical system of the two-cluster regulatory circuitry is more robust than the genetic circuit of two single TFs.ConclusionsWe propose that a modular hierarchy of regulatory circuits, each consisting of two mutually inhibiting and auto-activating TF clusters, can form hierarchical lineage bifurcations with improved safeguarding of critical early embryogenesis against biological perturbations. Furthermore, our computationally fast framework for modeling and visualizing the epigenetic landscape can be used to obtain insights from experimental data of development at the single cell resolution.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-015-0169-8) contains supplementary material, which is available to authorized users.

This paper proves that if G is a graph (parallel edges allowed) of maximum degree 3, then 0 c (G) 11=3 provided that G does not contain H 1 or H 2 as a subgraph, where H 1 and H 2 are obtained by subdividing one edge of K 3 2 (the graph with three parallel edges between two vertices) and K 4 , respectively. As

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