Total Unimodularity
A square, integer matrix B is unimodular (UM) if its determinant is 1 or -1.
An integer matrix A is called totally unimodular (TUM) if every square, nonsingular submatrix of A is UM.
From Cramer’s rule, it follows that if A is TUM and b is an integer vector, then every BFS of the constraint system Ax = b is integer.