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Transitive reductions and minimal equivalent subgraphs have proven to be a powerful concept to simplify networks and to measure their redundancy. Here we consider a generalization of the minimal equivalent subgraph problem where a set of arcs is already given. For two digraphs D = (V,A), D′ = (V,A′) with A′ ⊆ A, we ask for the minimal set of edges of D that have to be added to D′ such that the transitive closure of D equals the transitive closure of D′.
We present a method to compute such an extension and show that if D is transitively closed, this problem can be solved in polynomial time.
Flux coupling analysis (FCA) has become a useful tool for aiding metabolic reconstructions and guiding genetic manipulations. Originally, it was introduced for constraint-based models of metabolic networks that are based on the steady-state assumption. Recently, we have shown that the steady-state assumption can be replaced by a much weaker lattice-theoretic property related to the supports of metabolic fluxes. In this paper, we further extend our approach and delevelop an efficient algorithm for general qualitative flux coupling analysis (QFCA). We illustrate our method by thermodynamic flux coupling analysis (tFCA), which allows studying steady-state metabolic models with loop-law thermodynamic constraints. These models do not satisfy the lattice-theoretic properties required in our previous work. For a selection of genome-scale metabolic network reconstructions, we discuss both theoretically and practically, how thermodynamic constraints strengthen the coupling results that can be obtained with classical FCA.