并不是所有看起来简单的度序列问题都很容易
Not all simple looking degree sequence problems are easy
摘要：
度序列（DS）问题已经存在了至少一百二十年，随着网络科学的出现，越来越多的复杂的结构化DS问题被发明出来。有趣的是，到目前为止，所有这些问题在计算上都很容易。然而，很明显，我们很快就会发现计算困难的DS问题。在本文中，我们希望找到这样的困难DS问题与相对简单的定义。对于简单图$G$中的顶点$v$，用$d_i(v)$表示距离$v$恰好为$i$的顶点的数目。向量$\mathbf{d}^2(G)=( (d_1(v_1), d_2(v_1)), \ldots,$ $(d_1(v_n), d_2(v_n))$是图$G$的{\bf二阶度序列}.本文证明了判定自然数序列$((i_1,j_1),\ldots (i_n,j_n))$是否为简单无向图$G$的二阶度序列的问题是强NP-完全的.然后，我们将讨论一些进一步的NP完全DS问题。
英文摘要：
Degree sequence (DS) problems are around for at least hundred twenty years, and with the advent of network science, more and more complicated, structured DS problems were invented. Interestingly enough all those problems so far are computationally easy. It is clear, however, that we will find soon computationally hard DS problems. In this paper we want to find such hard DS problems with relatively simple definition. For a vertex $v$ in the simple graph $G$ denote $d_i(v)$ the number of vertices at distance exactly $i$ from $v$. Then $d_1(v)$ is the usual degree of vertex $v.$ The vector $\mathbf{d}^2(G)=( (d_1(v_1), d_2(v_1)), \ldots,$ $(d_1(v_n), d_2(v_n))$ is the {\bf second order degree sequence} of the graph $G$. In this note we show that the problem to decide whether a sequence of natural numbers $((i_1,j_1),\ldots (i_n,j_n))$ is a second order degree sequence of a simple undirected graph $G$ is strongly NP-complete. Then we will discuss some further NP-complete DS problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
