We identify a new decidable class of security protocols, both for reachability and equivalence properties. Our result holds for an unbounded number of sessions and for protocols with nonces. It covers all standard cryptographic primitives. Our class sets up three main assumptions. (i) Protocols need to be "simple", meaning that an attacker can precisely identify from which participant and which session a message originates from. We also consider protocols with no else branches (only positive test). (ii) Protocols should be type-compliant, which is intuitively guaranteed as soon as two encrypted messages of the protocol cannot be confused. (iii) Finally, we define the notion of a dependency graph, which, given a protocol, characterises how actions depend on the other ones (both sequential dependencies and data dependencies are taken into account). Whenever the graph is acyclic, then the protocol falls into our class. We show that many protocols of the literature belong to our decidable class, including for example some of the protocols embedded in the biometric passport.