Complete symmetry breaking constraints for the class of uniquely Hamiltonian graphs

This paper introduces, for the first time, a complete symmetry breaking constraint of polynomial size for a significant class of graphs: the class of uniquely Hamiltonian graphs. We introduce a canonical form for uniquely Hamiltonian graphs and prove that testing whether a given uniquely Hamiltonian graph is canonical can be performed efficiently. Based on this canonicity test, we construct a complete symmetry breaking constraint of polynomial size which is satisfied only by uniquely Hamiltonian graphs which are canonical. We apply the proposed symmetry breaking constraint to show new results regarding the class of uniquely Hamiltonian graphs. We also show that the proposed approach applies almost directly for the class of graphs which contain any cycle of known length where it shown to result in a partial symmetry breaking constraint. Given that it is unknown if there exist complete symmetry breaking constraints for graphs of polynomial size, this paper makes a first step in the direction of identifying specific classes of graphs for which such constraints do exist.