The completable digraphs for the totally nonnegative completion problem

In this paper, we study the totally nonnegative completion problem when the partial totally nonnegative matrix is non-combinatorially symmetric. In general, this type of partial matrix does not have a totally nonnegative completion. Here, we give necessary and sufficient conditions for completion of a partial totally nonnegative matrix to a totally nonnegative matrix in the cases where the digraph of the off-diagonal specified entries takes certain forms as path, cycles, alternate paths, block graphs, etc., distinguishing between the monotonically and non-monotonically labeled case.