Canonical Labelling of Site Graphs

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

A strand graph semantics for DNA-based computation

“…The canonical form is obtained by finding a canonical labelling of the nodes. In [32] a series of worst-case quadratic algorithms for finding a canonical labelling of site graphs is presented. There, a site graph is first transformed into a graph with coloured edges and then the ordering of the colours is used to define edge enumeration.…”

KaDE: A Tool to Compile Kappa Rules into (Reduced) ODE Models

“…From a combinatorial point of view, BioNetGen reasons on agents with multiple occurrences of equivalent sites, which may make the detection of embeddings exponentially costly (with respect to the number of agents). In constrast, KaDE quotients the set of biomolecular species on the fly: it reasons on rigid site graphs for which the detection of embeddings is at worst quadratic [26,27].…”