Characterizing the combinatorics of distributed EPR pairs for multi-partite entanglement

We develop protocols for preparing a GHZ state and, in general,a pure multi-partite maximally entangled state in a distributed network with apriori quantum entanglement between agents using classical communication and local operations. We investigate and characterize the minimal combinatorics of the sharing of EPR pairs required amongst agents in a network for the creation of multi-partite entanglement. We also characterize the minimal combinatorics of agents in the creation of pure maximal multi-partite entanglement amongst the set $N$ of $n$ agents in a network using apriori multi-partite entanglement states amongst subsets of $N$. We propose protocols for establishing multi-partite entanglement in the above cases.