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We study solutions to the multi-marginal Monge-Kantorovich problem which areconcentrated on several graphs over the first marginal. We first present twogeneral conditions on the cost function which ensure, respectively, that anysolution must concentrate on either finitely many or countably many graphs. Weshow that local differential conditions on the cost, known to imply local$d$-rectifiability of the solution, are sufficient to imply a local version ofthe first of our conditions. We exhibit two examples of cost functionssatisfying our conditions, including the Coulomb cost from density functionaltheory in one dimension. We also prove a number of results relating to theuniqueness and extremality of optimal measures. These include a sufficientcondition on a collection of graphs for any competitor in the Monge-Kantorovichproblem concentrated on them to be extremal, and a general negative result,which shows that when the problem is symmetric with respect to permutations ofthe variables, uniqueness cannot occur except under very special circumstances.

Solutions to multi-marginal optimal transport problems concentrated on several graphs