Multi-marginal optimal transport, a natural extension of the well-known classical optimal transport problem, is the problem of correlating given probability measures as efficiently as possible relative to a given cost function. Although a variety of applications have arisen over the past twelve years, the structure of solutions for the multi-marginal case has been difficult to address, mainly due to the strong dependence on the cost function. In this talk, I will briefly introduce the problem, summarize the known results for uniqueness in the multi-marginal case, and connect them with some past joint works with Brendan Pass based on costs associated with graphs.