Ternary graph isomorphism in polynomial time, after Luks

TL;DR

This paper presents an efficient implementation of Luks' polynomial-time algorithm for ternary graph isomorphism in the SAGE system, extending its application to rooted phylogenetic networks.

Contribution

The paper provides the first practical implementation of Luks' theoretical algorithm for ternary graphs and adapts it for phylogenetic networks.

Findings

Successful implementation of Luks' algorithm in SAGE

Efficient isomorphism testing for ternary graphs

Application to phylogenetic networks

Abstract

The graph isomorphism problem has a long history in mathematics and computer science, with applications in computational chemistry and biology, and it is believed to be neither solvable in polynomial time nor NP-complete. E. Luks proposed in 1982 the best algorithm so far for the solution of this problem, which moreover runs in polynomial time if an upper bound for the degrees of the nodes in the graphs is taken as a constant. Unfortunately, Luks' algorithm is purely theoretical, very difficult to use in practice, and, in particular, we have not been able to find any implementation of it in the literature. The main goal of this paper is to present an efficient implementation of this algorithm for ternary graphs in the SAGE system, as well as an adaptation to fully resolved rooted phylogenetic networks on a given set of taxa.