Solution to a Forcible Version of a Graphic Sequence Problem

Let $$A_n=(a_1,a_2,\ldots ,a_n)$$ and $$B_n=(b_1,b_2,\ldots ,b_n)$$ be nonnegative integer sequences with $$A_n\le B_n$$ . The purpose of this note is to give a good characterization such that every sequence $$\pi =(d_1,d_2,\ldots d_n)$$ even sum \pi \le graphic. This solves forcible version problem posed by Niessen generalizes the Erdős–Gallai theorem.