Elementary matrix equivalence and core transformation graphs for Parikh matrices

Abstract

The introduction of the Parikh matrix mapping by Mateescu et al. in 2001 gave rise to the injectivity problem, that is, the characterization of words having the same Parikh matrix. Certain elementary rewriting rules have been successful in solving this problem for the binary alphabet, yet they do not suffice for the general case. This paper studies these elementary rewriting rules exclusively for the ternary alphabet. Certain labeled graphs, termed as core transformation graphs, are introduced and used to study the structural changes in ternary words upon applications of these elementary rewriting rules. For an arbitrary core transformation graph, all possibilities of the labels on its edges are exhausted and, for each label, the pairs of vertices forming the corresponding edges are characterized. Finally, we obtain results relating core transformation graphs to elementarily matrix equivalent words.