  We study scenarios where multiple sellers of a homogeneous good compete on
prices, where each seller can only sell to some subset of the buyers.
Crucially, sellers cannot price-discriminate between buyers. We model the
structure of the competition by a graph (or hyper-graph), with nodes
representing the sellers and edges representing populations of buyers. We study
equilibria in the game between the sellers, prove that they always exist, and
present various structural, quantitative, and computational results about them.
We also analyze the equilibria completely for a few cases. Many questions are
left open.
