Let G be the node-arc incidence matrix of a given directed network (rows of $ G $ correspond to nodes and their columns correspond to arcs. To let $ B_1, dots, B_K $ designate a partition of the nodes of the network. Suppose the network is such that a directed arc can go in from a node $ B_k $ to another node in $ B_ ell $ only if $ k < ell $, To let $ H $ denote a matrix with $ K $ Rows and assume that its columns are indexed by the arcs of the underlying network. We assume that $ H $ is that his $ (k, e) $-th entry is equal to one $ e in B_k $and nothing else.
Is the matrix (H; G) (obtained by the concatenation of rows of $ H $ and $ G $) completely unimodular? If not, can you give a counterexample?
I numerically examined some examples and confirmed the complete unimodularity for these examples. I thought it might be possible to exploit the structure of $ H $ (Note the special line structure) to formally prove the result. I tried to use the Ghouila Houri condition (see https://en.wikipedia.org/wiki/Unimodular_matrix), which appears to be a good candidate for exploiting the line structure. So far I have not been successful.