Multi-party contract signing (MPCS) protocols allow a group of signers to exchange signatures on a predefined contract. Previous approaches considered either completely linear protocols or fully parallel broadcasting protocols. We introduce the new class of DAG MPCS protocols which combines parallel and linear execution and allows for parallelism even within a signer role. This generalization is useful in practical applications where the set of signers has a hierarchical structure, such as chaining of service level agreements and subcontracting. Our novel DAG MPCS protocols are represented by directed acyclic graphs and equipped with a labeled transition system semantics. We define the notion of abort-chaining sequences and prove that a DAG MPCS protocol satisfies fairness if and only if it does not have an abort-chaining sequence. We exhibit several examples of optimistic fair DAG MPCS protocols. The fairness of these protocols follows from our theory and has additionally been verified with our automated tool. We define two complexity measures for DAG MPCS protocols, related to execution time and total number of messages exchanged. We prove lower bounds for fair DAG MPCS protocols in terms of these measures.