Linear time canonicalization and enumeration of non-Isomorphic 1-Face embeddings

Antiparallel strong traces (ASTs) are a type of walks in graphs which use every edge exactly twice. They correspond to 1-face embeddings in orientable surfaces and can be used to design self-assembling protein or DNA strands. Based on a novel canonical form invariant for ASTs, gap vector, we provide a linear-time isomorphism test for ASTs and thus, also for orientable 1-face embeddings of graphs. Using the canonical form, we develop an algorithm for enumerating all pairwise non-isomorphic 1-face embeddings of graphs. We compare our algorithm with an independent implementation of a recent algebraic approach (Bašić et al., MATCH Commun. Math. Comput. Chem. 78 (3), 2017) on large data sets. Our results yield the first large-scale enumeration of non-isomorphic embeddings and investigation of their properties.