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

Uniquely Pressable Graphs: Characterization, Enumeration, and Recognition