An efficient algorithm to test forcibly-biconnectedness of graphical degree sequences

TL;DR

This paper introduces a more efficient algorithm for testing forcibly biconnectedness in graphical degree sequences, improving over previous methods, and explores enumerative properties and conjectures related to these sequences.

Contribution

It presents a new algorithm with better average efficiency for testing forcibly biconnectedness and integrates it into enumeration algorithms to derive new asymptotic conjectures.

Findings

The algorithm is efficient on average despite exponential worst-case complexity.

Enumeration results suggest asymptotic proportions of certain forcibly biconnected sequences.

Conjectures about the limiting behavior of these proportions as sequence length grows.

Abstract

We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected or not and prove its correctness. The worst case run time complexity of the algorithm is shown to be exponential but still much better than the previous basic algorithm presented in \cite{Wang2018}. We show through experimental evaluations that the algorithm is efficient on average. We also adapt Ruskey et al's classic algorithm to enumerate zero-free graphical degree sequences of length $n$ and Barnes and Savage's classic algorithm to enumerate graphical partitions of an even integer $n$ by incorporating our testing algorithm into theirs and then obtain some enumerative results about forcibly biconnected graphical degree sequences of given length $n$ and forcibly biconnected graphical partitions of given even integer $n$. Based on these enumerative results we make some conjectures such as: when $n$ is large, (1) the proportion of forcibly biconnected graphical degree sequences of length $n$ among all zero-free graphical degree sequences of length $n$ is asymptotically a constant between 0 and 1; (2) the proportion of forcibly biconnected graphical partitions of even $n$ among all forcibly connected graphical partitions of $n$ is asymptotically 0.