The Sortability of Graphs and Matrices under Context Directed Swaps

TL;DR

This paper explores the concept of sorting graphs and matrices using constrained block interchange operations, providing enumeration formulas and asymptotic analysis for the class of sortable graphs.

Contribution

It generalizes the constrained block interchange operation to graphs and matrices, offering enumeration and asymptotic results for sortable graphs.

Findings

Closed-form enumeration of sortable graphs on n vertices.

Asymptotic proportion of sortable graphs as n grows large.

Extension of block interchange concepts to matrices and graphs.

Abstract

The study of sorting permutations by block interchanges has recently been stimulated by a phenomenon observed in the genome maintenance of certain ciliate species. The result was the identification of a block interchange operation that applies only under certain constraints. Interestingly, this constrained block interchange operation can be generalized naturally to simple graphs and to an operation on square matrices. This more general context provides numerous techniques applicable to the original context. In this paper we consider the more general context, and obtain an enumeration, in closed form, of all simple graphs on n vertices that are ``sortable" by the graph analogue of the constrained version of block interchanges. We also obtain asymptotic results on the proportion of graphs on n vertices that are so sortable.