Not all simple looking degree sequence problems are easy

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 d 2 (G) = ((d 1 (v 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 ((i 1 , j 1 ), . . . (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. Primary 05C07, 60K35; secondary 68R05.

Constructing bounded degree graphs with prescribed degree and neighbor degree sequences

“…This motivated the question: "What kind of local properties make the graph construction hard?" Erdős and Miklós showed that it is already NP-complete to ask if there exists a graph with given degree and neighbor degree sequence [7].…”

“…is the degree of vertex v and N (v) is the set of the neighbors of v? Erdős and Miklós showed that this decision problem is NP-complete in general [7]. Here we consider a specific case: the maximum degree is 4, and we are looking for connected realizations.…”