I read this statement related to Havel and Hakimi.
Suppose that $\textbf{d}=(d_1,d_2,\ldots,d_n)$ be a nonincreasing sequence of nonnegative integers. Let \textbf{d'}=(d_2-1,d_3-1,\ldots, d_{d_1+1}-1,d_{d_1+2},\ldots,d_n). Then $\textbf{d}$ is graphical if and only if \textbf{d'} is graphical.
Why is this the case? For suppose $G$ is the graph associated to the first sequence. It seems the the second sequence is obtained by removing the vertex $v_1$ from $G$, which induces a subgraph by deleting all $d_1$ edges adjacent to $v_1$, and this would decrease the degree of $d_1$ vertices, giving a graph G' with the second degree sequence. But this argument seems to assume that $v_1$ is connected to the next $d_1$ vertices of highest degree. What is $v_1$ is adjacent to $v_n$ or any $v_k$ for any $k>{d_1+1}$? Why would the claim still be true? Thank you.