Structure-complex systems with threshold survival

Earlier there was obtained a characterization of complex systems modeled by $K$-terminal undirected networks with threshold survival. The characterization of complex systems modelled by $K$-terminal directed networks with threshold survival was an open problem. The solution of this problem directly follows from the characterization of $dc$-trivial graphs with threshold survival given in the paper. The $dc$-trivial graphs form a subset of monotone graphs and include as special cases all classic multiterminal reliability networks. The general class of monotone graphs is proved to be recognized in time polynomial in the number of their minpaths.