Summary: We study cellular automata with additive rules on finite undirected graphs. The addition is carried out in some finite abelian monoid. We investigate the problem of deciding whether a given configuration has a predecessor. Depending on the underlying monoid this problem is solvable in polynomial time or NP-complete. Furthermore, we study the global reversibility of cellular graph automata based on addition modulo two. We give a linear time algorithm to decide reversibility of unicyclic graphs.