Solving a special case of the P conjecture using dependency graphs with dissolution

We solve affirmatively a new special case of the P conjecture by Gh. Păun, which states that P systems with active membranes without charges and without non-elementary membrane division cannot solve -complete problems in polynomial time. The variant we consider is monodirectional, i.e., without send-in communication rules, shallow, i.e., with membrane structures consisting of only one level besides the external membrane, and deterministic, rather than more generally confluent. We describe a polynomial-time Turing machine simulation of this variant of P systems, exploiting a generalised version of dependency graphs for P systems which, unlike the original version introduced by Cordón-Franco et al., also takes membrane dissolution into account.