Breaking Symmetries in Graph Search with Canonizing Sets

TL;DR

This paper presents a method for efficiently breaking symmetries in small graph search problems, enabling enumeration of all non-isomorphic solutions and extending graph enumeration sequences.

Contribution

It introduces complete, compact symmetry breaking constraints for small graph searches, including instance-dependent constraints for larger problems, and generalizes to matrix search problems.

Findings

Enumerates all non-isomorphic solutions for small graphs.

Computes instance-dependent symmetry breaking constraints.

Extends known graph enumeration sequences from OEIS.

Abstract

There are many complex combinatorial problems which involve searching for an undirected graph satisfying given constraints. Such problems are often highly challenging because of the large number of isomorphic representations of their solutions. This paper introduces effective and compact, complete symmetry breaking constraints for small graph search. Enumerating with these symmetry breaks generates all and only non-isomorphic solutions. For small search problems, with up to $10$ vertices, we compute instance independent symmetry breaking constraints. For small search problems with a larger number of vertices we demonstrate the computation of instance dependent constraints which are complete. We illustrate the application of complete symmetry breaking constraints to extend two known sequences from the OEIS related to graph enumeration. We also demonstrate the application of a generalization of our approach to fully-interchangeable matrix search problems.