Degree-based graph construction

Abstract

Degree-based graph construction is an ubiquitous problem in network modeling, ranging from social sciences to chemical compounds and biochemical reaction networks in the cell. This problem includes existence, enumeration, exhaustive construction and sampling questions with aspects that are still open today. Here we give necessary and sufficient conditions for a sequence of nonnegative integers to be realized as a simple graph's degree sequence, such that a given (but otherwise arbitrary) set of connections from a arbitrarily given node are avoided. We then use this result to present a swap-free algorithm that builds {\em all} simple graphs realizing a given degree sequence. In a wider context, we show that our result provides a greedy construction method to build all the $f$-factor subgraphs embedded within $K_n\setminus S_k$, where $K_n$ is the complete graph and $S_k$ is a star graph centered on one of the nodes.