String and band complexes over SAG algebras

We give a combinatorial description of a family of indecomposable objects in the bounded derived categories of string almost gentle algebras. These indecomposable objects are, up to isomorphism, the string and band complexes introduced by V. Bekkert and H. Merklen. With this description, we give a sufficient condition for a given string complex to have infinite minimal projective resolution, and we extend this condition for the case of string algebras.