Not all simple looking degree sequence problems are easy

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.