The notion of a 2-Segal object was recently defined by Dyckerhoff and Kapranov, and independently by Gálvez-Carrillo, Kock, and Tonks under the name of decomposition space.  Whereas 1-Segal sets model the structure of a category, in which composition is defined and is associative, 2-Segal sets instead encode a more general structure in which composition need not exist or be unique, but is still associative when it is defined.   The 2-Segal set associated to a graph gives a nice example where maps can be composed in different ways. In particular, following a definition of Dyckerhoff and Kapranov, this 2-Segal set has an associated Hall algebra which is much smaller than most natural examples of such algebras and has a curious description as a cohomology ring.