Simply-laced mixed-sign Coxeter groups, with an associate graph is a line or a simple cycle

In 2011 Eriko Hironaka introduced an interesting generalization of Coxeter groups, motivated by studying certain mapping classes. The generalization is by labeling the vertices of a Coxeter graph either by +1 or by -1, and then generalizing the standard geometric representation of the associated Coxeter group by concerning the labels of the vertices. The group which Hironaka get by that generalization is called mixed-sign Coxeter group. In this paper we classify the simply-laced mixed-sign Coxeter groups where the associated graph is either a line or a simple cycle. We show that all the defining relations of the mixed-sign Coxeter groups with the mentioned associated graph (either a line or a simple cycle) are squares or cubes of a product of conjugates of two generators of the mixed-sign Coxeter group and are strongly connected to the labels of the vertices of the associated graph.