A simplification of the C-realizability criterion for the Nonnegative Inverse Eigenvalue Problem for integers

A multiset $\mathrm{\Lambda }=\{{\lambda }_1,\dots ,{\lambda }_n\}$ of complex numbers is said to be realizable whenever there exists a nonnegative matrix of order $n$ with spectrum $\mathrm{\Lambda }$. One of the broadest criterion that guarantees realizability is the $C-$realizability. It says that $\mathrm{\Lambda }$, with real numbers, is $C-$realizable if it can be obtained starting from $n$ basic multisets $\{0\},\dots ,\{0\}$ by successively applying any finite number of times any of the following rules: (a) join two of the multisets; (b) increase by $\epsilon >0$ the Perron root of one of the multisets; © increase by $\epsilon >0$ the Perron root of one of the multisets and simultaneously increase or decrease by $\epsilon$ any other value of the same multiset.