Efficient Counting of Degree Sequences

TL;DR

This paper introduces efficient polynomial-time algorithms for counting specific classes of degree sequences, enabling the enumeration of these sequences for larger n and providing evidence for a mathematical conjecture.

Contribution

The paper presents the first polynomial-time algorithms for counting zero-free, connected, and biconnected degree sequences, significantly improving computational efficiency.

Findings

Able to tabulate degree sequences up to n=118

Supports a conjecture about the limit of |D(n)|/|D(n-1)|

Provides new numerical data for known OEIS sequences

Abstract

Novel dynamic programming algorithms to count the set $D(n)$ of zero-free degree sequences of length $n$, the set $D_c(n)$ of degree sequences of connected graphs on $n$ vertices and the set $D_b(n)$ of degree sequences of biconnected graphs on $n$ vertices exactly are presented. They are all based on a recurrence of Barnes and Savage and shown to run in polynomial time and are asymptotically much faster than the previous best known algorithms for these problems. These appear to be the first polynomial time algorithms to compute $|D(n)|$, $|D_c(n)|$ and $|D_b(n)|$ to the author's knowledge and have enabled us to tabulate them up to $n=118$, the majority of which were unknown. The available numerical results of $|D(n)|$ tend to give more supporting evidence of a conjecture of Gordon F. Royle about the limit of $|D(n)|/|D(n-1)|$. The OEIS entries that can be computed by algorithms in this paper are A004251, A007721, A007722 and A095268.