An assembly problem refers to a situation where a buyer wants to purchase a fixed number of complementary items from sellers holding an item each. We model complementarity using graphs where nodes represent items, and edges between two nodes represent a complementary relationship between these items. The buyer wants to purchase a feasible path in the graph, i.e., a path of desired length, where the sum of valuations of the sellers owning the items do not exceed buyer’s own valuation. A seller is critical if he lies on every feasible path. We examine subgame perfect equilibria of an infinite horizon alternate-offer bargaining game between the buyer and the sellers. We show that there exist equilibria where the buyer can extract full surplus within two periods if and only if (a) there are no critical sellers and (b) there exist at least two feasible paths with minimum sum of seller valuations. We also characterize the upper bounds on buyer’s surplus when she cannot extract full surplus. Thus we characterize the trade-off between complementarity and competition in terms of buyer’s equilibrium surplus share in assembly problems.