Constructing new realisable lists from old in the NIEP

Given a list of complex numbers σ:=(λ1,λ2,…,λn), we say that σ is realisable if σ is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (or NIEP) is the problem of characterising all realisable lists.Given a realisable list (ρ,λ2,λ3,…,λm), where ρ is the Perron eigenvalue and λ2 is real, we find families of lists (μ1,μ2,…,μn), for which(μ1,μ2,…,μn,λ3,λ4,…,λm) is realisable. In addition, given a realisable list(ρ,α+iβ,α−iβ,λ4,λ5,…,λm), where ρ is the Perron eigenvalue and α and β are real, we find families of lists (μ1,μ2,μ3,μ4), for which (μ1,μ2,μ3,μ4,λ4,λ5,…,λm) is realisable.