Let (a₁,a₂,…,a_n) and (b₁,b₂,…,b_n) be two sequences of nonnegative integers satisfying the condition that b₁≥b₂≥⋯≥b_n, a_i≤ b_i for i=1,2,…,n and a_i+b_i≥a_i+1+b_i+1 for i=1,2,…, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a₁,a₂,…,a_n) and (b₁,b₂,…,b_n) such that for every (c₁,c₂,…,c_n) with a_i≤c_i≤b_i for i=1,2,…,n and ∑&limits;_i=1ⁿ c_i≡0 (mod 2), there exists a simple graph G with vertices v₁,v₂,…,v_n such that d_G(v_i)=c_i for i=1,2,…,n. This is a variant of Niessen's problem on degree sequences of graphs (Discrete Math., 191 (1998), 247 13;253).