Graph reductions, binary rank, and pivots in gene assembly

TL;DR

This paper introduces a generalized graph reduction framework related to gene assembly, analyzing its properties through adjacency matrices over GF(2), and connects it to matrix pivot operations, resolving open problems.

Contribution

It generalizes gene assembly graph reductions, studies their invariance and properties via binary rank, and links them to matrix pivot operations, solving previously open questions.

Findings

Graph reductions are path invariant, depending only on vertices removed.

Binary rank determines the number of negative rule applications in reductions.

Graph reductions are closely related to matrix pivot operations.

Abstract

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.