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We describe a graph reduction operation, generalizing three graph reductionoperations related to gene assembly in ciliates. The graph formalization ofgene assembly considers three reduction rules, called the positive rule, doublerule, and negative rule, each of which removes one or two vertices from agraph. The graph reductions we define consist precisely of all compositions ofthese rules. We study graph reductions in terms of the adjacency matrix of agraph over the finite field with two elements, and show that they are pathinvariant, in the sense that the result of a sequence of graph reductionsdepends only on the vertices removed. The binary rank of a graph is the rank ofits adjacency matrix over the finite field with two elements. We show that thebinary rank of a graph determines how many times the negative rule is appliedin any sequence of positive, double, and negative rules reducing the graph tothe empty graph, resolving two open problems posed by Harju, Li, and Petre. Wealso demonstrate the close relation between graph reductions and the matrixpivot operation, both of which can be studied in terms of the poset of subsetsof vertices of a graph that can be removed by a graph reduction.

Graph reductions, binary rank, and pivots in gene assembly