Canonical Labelling of Site Graphs

TL;DR

This paper explores algorithms for canonical labelling of site graphs, reducing the problem to edge-coloured graph labelling, and presents multiple algorithms with quadratic worst-case complexity, optimized for graphs with many automorphisms or few bisimulation equivalences.

Contribution

It introduces three new canonical labelling algorithms for site graphs, including extensions of existing algorithms, with sub-quadratic performance in specific graph categories.

Findings

Algorithms run in quadratic worst-case time.

One algorithm is sub-quadratic for graphs with many automorphisms.

Another is sub-quadratic for graphs with few bisimulation equivalences.

Abstract

We investigate algorithms for canonical labelling of site graphs, i.e. graphs in which edges bind vertices on sites with locally unique names. We first show that the problem of canonical labelling of site graphs reduces to the problem of canonical labelling of graphs with edge colourings. We then present two canonical labelling algorithms based on edge enumeration, and a third based on an extension of Hopcroft's partition refinement algorithm. All run in quadratic worst case time individually. However, one of the edge enumeration algorithms runs in sub-quadratic time for graphs with "many" automorphisms, and the partition refinement algorithm runs in sub-quadratic time for graphs with "few" bisimulation equivalences. This suite of algorithms was chosen based on the expectation that graphs fall in one of those two categories. If that is the case, a combined algorithm runs in sub-quadratic worst case time. Whether this expectation is reasonable remains an interesting open problem.