Degree-based graph construction

TL;DR

This paper establishes conditions for constructing simple graphs with specified degree sequences while avoiding certain connections, and introduces a swap-free algorithm to generate all such graphs, with applications to subgraph embedding.

Contribution

It provides necessary and sufficient conditions for degree sequence realizability with connection constraints and presents a novel swap-free graph construction algorithm.

Findings

Derived conditions for degree sequence realizability with connection constraints

Developed a swap-free algorithm to generate all graphs with a given degree sequence

Applied results to construct all f-factor subgraphs within a complete graph minus a star

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.