# Uniquely Pressable Graphs: Characterization, Enumeration, and Recognition

## TL;DR

This paper characterizes, counts, and develops a recognition algorithm for uniquely pressable graphs, which have a single pressing sequence transforming them into empty graphs, with applications in phylogenetics.

## Contribution

It provides a complete characterization, enumeration, and a polynomial-time recognition algorithm for uniquely pressable graphs, addressing a question from prior research.

## Key findings

- Characterization of uniquely pressable graphs
- Counting of such graphs on a given number of vertices
- Polynomial-time recognition algorithm

## Abstract

We consider "pressing sequences", a certain kind of transformation of graphs with loops into empty graphs, motivated by an application in phylogenetics. In particular, we address the question of when a graph has precisely one such pressing sequence, thus answering an question from Cooper and Davis (2015). We characterize uniquely pressable graphs, count the number of them on a given number of vertices, and provide a polynomial time recognition algorithm. We conclude with a few open questions. Keywords: Pressing sequence, adjacency matrix, Cholesky factorization, binary matrix