Characterization of graph classes by forbidden structures and reductions

We have shown in this paper, that forbidden structures and reductions describe different classes of graph languages. For important applications, the graph language under consideration can be defined by both concepts. It is an open problem, how to characterize such languages. It would be very useful to restrict systems of forbidden structures such that one can to construct an equivalent reduction system. In (3.2), a reduction system with NP-complete membership-problem is given. The membership -problem remains NP-complete even, if the transformations T and T1 are removed. It is an open question, whether there is a reduction system with one rule and NP-complete membership-problem. Furthermore, reduction systems should be restricted such that the membership-problem is solvable in polynomial time. Obviously, the required property should be decidable.