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

TL;DR

This paper introduces an efficient algorithm to determine if a graphical degree sequence is forcibly connected, extends it to k-connectedness, and provides enumerative insights and conjectures about the prevalence of forcibly connected sequences.

Contribution

The paper presents a novel algorithm for testing forcibly connected graphical degree sequences and extends it to k-connectedness, along with enumeration and conjectural analysis.

Findings

Algorithm is efficient on average but may have exponential worst-case runtime.

Most zero-free graphical degree sequences are forcibly connected for large n.

Few graphical partitions of even n are forcibly connected for large n.

Abstract

We present an algorithm to test whether a given graphical degree sequence is forcibly connected or not and prove its correctness. We also outline the extensions of the algorithm to test whether a given graphical degree sequence is forcibly $k$-connected or not for every fixed $k\ge 2$. We show through experimental evaluations that the algorithm is efficient on average, though its worst case run time is probably exponential. 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 even integer $n$ by incorporating our testing algorithm into theirs and then obtain some enumerative results about forcibly connected graphical degree sequences of given length $n$ and forcibly connected graphical partitions of given even integer $n$. Based on these enumerative results we make some conjectures such as: when $n$ is large, (1) almost all zero-free graphical degree sequences of length $n$ are forcibly connected; (2) almost none of the graphical partitions of even $n$ are forcibly connected.