  We describe a graph reduction operation, generalizing three graph reduction
operations related to gene assembly in ciliates. The graph formalization of
gene assembly considers three reduction rules, called the positive rule, double
rule, and negative rule, each of which removes one or two vertices from a
graph. The graph reductions we define consist precisely of all compositions of
these rules. We study graph reductions in terms of the adjacency matrix of a
graph over the finite field with two elements, and show that they are path
invariant, in the sense that the result of a sequence of graph reductions
depends only on the vertices removed. The binary rank of a graph is the rank of
its adjacency matrix over the finite field with two elements. We show that the
binary rank of a graph determines how many times the negative rule is applied
in any sequence of positive, double, and negative rules reducing the graph to
the empty graph, resolving two open problems posed by Harju, Li, and Petre. We
also demonstrate the close relation between graph reductions and the matrix
pivot operation, both of which can be studied in terms of the poset of subsets
of vertices of a graph that can be removed by a graph reduction.
