Practical Canonical Labeling of Multi-Digraphs via Computer Algebra

Practical algorithms for computing canonical forms of multi-digraphs do not exist in the literature. This paper proposes two practical approaches for finding canonical forms, from the perspective of nD symbolic computation. Initially, the approaches turn the problem of finding canonical forms of multi-digraphs into computing canonical forms of indexed monomials in computer algebra. Then, the first approach utilizes the double coset representative method in computational group theory for canonicalization of indexed monomials and shows that finding the canonical forms of a class of multi-digraphs in practice has polynomial complexity of approximately O((k+p)2) or O(k2.1) by the computer algebra system (CAS) tool Tensor-canonicalizer. The second approach verifies the equivalence of canonicalization of indexed monomials and finding canonical forms of (simple) colored tripartite graphs. It is found that the proposed algorithm takes approximately O((k+2p)4.803) time for a class of multi-digraphs in practical implementation, combined with one of the best known graph isomorphism tools Traces, where k and p are the vertex number and edge number of a multi-digraph, respectively.