Ideal bases in constructions defined by directed graphs

The present article continues the investigation of visible ideal bases in constructions defined using directed graphs. Our main theorem establishes that, for every balanced digraph D and each idempotent semiring R with 1, the incidence semiring ID(R) of the digraph D has a convenient visible ideal basis BD(R). It also shows that the elements of BD(R) can always be used to generate two-sided ideals with the largest possible weight among the weights of all two-sided ideals in the incidence semiring.