In this paper, we consider the problem of inferring a Boolean network (BN) from a given set of singleton attractors, where it is required that the resulting BN has the same set of singleton attractors as the given one. We show that the problem can be solved in linear time if the number of singleton attractors is at most two and each Boolean function is restricted to be a conjunction or disjunction of literals. We also show that the problem can be solved in polynomial time if more general Boolean functions can be used. In addition to the inference problem, we study two network completion problems from a given set of singleton attractors: adding the minimum number of edges to a given network, and determining Boolean functions to all nodes when only network structure of a BN is given. In particular, we show that the latter problem cannot be solved in polynomial time unless P=NP, by means of a polynomial-time Turing reduction from the complement of the another solution problem for the Boolean satisfiability problem.

The impact degree is a measure of the robustness of a metabolic network against deletion of single or multiple reaction(s). Although such a measure is useful for mining important enzymes/genes, it was defined only for networks without cycles. In this paper, we extend the impact degree for metabolic networks containing cycles and develop a simple algorithm to calculate the impact degree. Furthermore we improve this algorithm to reduce computation time for the impact degree by deletions of multiple reactions. We applied our method to the metabolic network of E. coli, that includes reference pathways, consisting of 3281 reaction nodes and 2444 compound nodes, downloaded from KEGG database, and calculate the distribution of the impact degree. The results of our computational experiments show that the improved algorithm is 18.4 times faster than the simple algorithm for deletion of reaction-pairs and 11.4 times faster for deletion of reaction-triplets. We also enumerate genes with high impact degrees for single and multiple reaction deletions.