Not all simple looking degree sequence problems are easy

TL;DR

This paper demonstrates that determining whether a given sequence can be realized as a second order degree sequence of a simple graph is strongly NP-complete, revealing computational hardness in seemingly simple degree sequence problems.

Contribution

It introduces the second order degree sequence problem and proves its strong NP-completeness, highlighting complexity in structured degree sequence problems.

Findings

Second order degree sequence recognition is strongly NP-complete.

Many simple-looking degree sequence problems are computationally hard.

Discussion of additional NP-complete degree sequence problems.

Abstract

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.