Obviously, global indicates local, and local indicates pairwise. However, the reverse is not necessary true. However, for a distribution p(y) > 0, these three properties are equivalent. Proof can be found from Koller's book.

One Problem: when can we make use of these three properties? Answer: during graph construction.

In most computer vision applications, graph structure is not learnt from data, but constructed from specific applications. So there is no graph construction step in most CV problems. However, in data mining, the underlying graph structure of the big data is hidden from us, which is not easy to get. Then we have to learn structure from data. At this time, it's more convenient to check whether two nodes are independent than to check whether a bunch of nodes A are independent from another bunch of nodes B. If we further assume p(y) > 0 for any y, then from pairwise independence, we get global independence easily with no need to probablistically checking.