Successful Pressing Sequences for a Bicolored Graph and Binary Matrices

TL;DR

This paper explores successful pressing sequences in bicolored graphs using matrix theory over GF(2), providing new characterizations, bounds, and connections to computational phylogenetics.

Contribution

It introduces novel linear-algebraic and graph-theoretic characterizations of pressing sequences and relates them to permutation sortings, expanding understanding in graph theory and phylogenetics.

Findings

Multiple alternative characterizations of successful pressing sequences.

Bounds established on the number of such sequences.

Connections made between pressing sequences and permutation sortings.

Abstract

We apply matrix theory over $\mathbb{F}_2$ to understand the nature of so-called "successful pressing sequences" of black-and-white vertex-colored graphs. These sequences arise in computational phylogenetics, where, by a celebrated result of Hannenhalli and Pevzner, the space of sortings-by-reversal of a signed permutation can be described by pressing sequences. In particular, we offer several alternative linear-algebraic and graph-theoretic characterizations of successful pressing sequences, describe the relation between such sequences, and provide bounds on the number of them. We also offer several open problems that arose as a result of the present work.