The second order degree sequence problem is NP-complete

The classical degree sequence problems for simple graphs are computationally easy. In the last years numerous new construction problems were introduced (for example the joint degree matrix problems and its variants) which are still easy. Network scientists would like to generate networks with more and more constraints and it is expected that the complexity of constructing networks with prescribed constraints turns from easy to hard as the amount of constraints are increased. In this paper we show a somewhat surprising result that some second order degree sequence problems are already NP-complete. For a vertex v in the simple graph G denote di(v) the number of vertices at distance exactly i from v. Then d1(v) is the usual degree of vertex v. The vector d2(G) = ((d1(v1), d2(v1)), . . . , (d1(vn), d2(vn)) is the second order degree sequence of the graph G. In this note we show that the problem to decide whether a sequence of natural numbers ((i1, j1), . . . (in, jn)) is a second order degree sequence of a simple undirected graph G is strongly NPcomplete. Similarly, denote D2(v) the sum of the degrees of the neighbors of v. (This is always bigger than d2(v) and several vertices may occur in it multiple times.) We also show that the decision problem whether a pair (d(G),D2(v)) is graphical is NP-complete.