Exact polynomial-time isomorphism testing in directed graphs through comparison of vertex signatures in Krylov subspaces.

Abstract

MotivationThe complexity of the isomorphism problem in directed graphs has remained unsolved to date. This study examines the properties of Krylov matrices, demonstrating that they may be used to generate vertex "signatures" to allow exact analogy testing in directed graphs. ResultsA "vertex signature" is defined by initialising a Krylov matrix with a binary vector indicating the vertex position. This study demonstrates that signatures of analogous vertices are related by a linear-ordering transformation. It is demonstrated that equality of ordered vertex signatures is necessary and sufficient to demonstrate analogy. Thus, analogous vertices may be identified by checking each of the n candidates sequentially. This result is extended to analogous vertex sets. Thus, the isomorphic mapping may be constructed iteratively [order](n5) time by building a set of vertex analogies sequentially. The algorithm is applied to a dataset of enzyme structures, with comparison to a common heuristic algorithm. Availability and ImplementationSource code is provided at github.com/robertoshea/graph_isomorphism_directed. Contactrobert.1.oshea@kcl.ac.uk Supplementary DataSupplementary results are attached.