 Total unimodularity of directed incidence matrices

The directed vertex-edge incidence matrix has one $+1$ and one $-1$ per nonloop column and a zero column for a loop. Every square subdeterminant belongs to $\{0,1,-1\}$. Induct on its size: a column with no nonzero entries makes the determinant zero; a column with one nonzero entry permits a cofactor expansion to a smaller determinant; and if every column has two nonzero entries, their opposite signs make the column sums zero, giving linearly dependent rows. Negating rows preserves absolute subdeterminants. Consequently the unsigned vertex-edge incidence matrix of a bipartite graph is totally unimodular: negate all rows in one part. The assignment problem is an instance, with row vertices and column vertices forming the two parts.