The derived Hall algebra of a graded gentle one-cycle algebra I: the triangle structure

Under a mild condition, the perfect derived category and the finite-dimensional derived category of a graded gentle one-cycle algebra are described as twisted root categories of certain infinite quivers of type $\mathbb{A}_\infty^\infty$. As a consequence, it is shown that the triangle structure of such derived categories is uniquely determined by the underlying additive category.