  An arithmetical structure on a finite, connected graph without loops is given
by an assignment of positive integers to the vertices such that, at each
vertex, the integer there is a divisor of the sum of the integers at adjacent
vertices, counted with multiplicity if the graph is not simple. Associated to
each arithmetical structure is a finite abelian group known as its critical
group. Keyes and Reiter gave an operation that takes in an arithmetical
structure on a finite, connected graph without loops and produces an
arithmetical structure on a graph with one fewer vertex. We study how this
operation transforms critical groups. We bound the order and the invariant
factors of the resulting critical group in terms of the original arithmetical
structure and critical group. When the original graph is simple, we determine
the resulting critical group exactly.
