  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
