Structure of words with short 2-length in a free product of groups

Howie and Duncan observed that a word in a free product with length at least two, which is not a proper power and involves no letter of order two can be decomposed as a product of two cyclic subwords each of which is uniquely positioned. Using this property, they proved various important results about a one-relator product of groups with such word as the relator. In this paper, we show that similar results hold in a more general setting where we allow a certain number of elements of order two. (C) 2018 Elsevier Inc. All rights reserved.