For a loopless, acyclic, transitive directed graph, we give a new result about the relation between the t-predecessor property of subsets of its levels and the property of having full combinatorial column rank for certain blocks in its adjacency matrix. As a corollary, we give new lower bounds for the ith height characteristic number of a singular M-matrix. An open question about determining the height characteristics of singular M-matrices with a prescribed singular graph is also posed.
