Insertions Yielding Equivalent Double Occurrence Words

A double occurrence word (DOW) is a word in which every symbol appears exactly twice; two DOWs are equivalent if one is a symbol-to-symbol image of the other. We consider the so called repeat pattern (αα) and the return pattern (ααR), with gaps allowed between the α’s. These patterns generalize square and palindromic factors of DOWs, respectively. We introduce a notion of inserting repeat/return words into DOWs and study how two distinct insertions into the same word can produce equivalent DOWs. Given a DOW w, we characterize the structure of w which allows two distinct insertions to yield equivalent DOWs. This characterization depends on the locations of the insertions and on the length of the inserted repeat/return words and implies that when one inserted word is a repeat word and the other is a return word, then both words must be trivial (i.e., have only one symbol). The characterization also introduces a method to generate families of words recursively