Bipartite Graphs and Perfect Matchings

When we think about markets creating opportunities for interaction between buyers and sellers, there is an implicit network encoding the access between buyers and sellers. In fact, there are several ways to use networks to model buyer-seller interaction, and here we discuss some of them. First, consider the case in which not all buyers have access to all sellers. There could be several reasons for this lack of access – it could be informational (certain sellers and buyers are not aware of each other), or institutional (regulations or conventions prohibit certain sellers from transacting with certain buyers), or, as we will see later, each buyer could prioritize the sellers somehow, and only be interested in trading with the highest-priority seller(s). One way to encode the pattern of potential interaction is as follows: we create a node for each seller and a node for each buyer, and we draw an edge between a seller and a buyer if they have the potential to engage in a transaction. We call such a graph a bipartite graph, since its nodes can be divided into two parts (in this case the sellers and buyers), in such a way that each edge has one endpoint in each part. Figure 1(a) gives some examples of bipartite graphs on sets of sellers and buyers; bipartite graphs are often drawn as in this figure, with the nodes in the two parts arranged in parallel columns. One of the most basic questions about bipartite graphs concerns the notion of a perfect matching. Suppose we have an equal number of sellers and buyers. We can ask: is it possible to pair up the sellers and buyers in such a way that each pair has the opportunity to trade (i.e. the two nodes in each pair are joined by an edge)? We call such a pairing a perfect matching (since all nodes are perfectly matched together, with no one left out). For our discussion below, it will be more useful to think of a perfect matching in the following equivalent way: it's a set of edges with the property that each node is the endpoint of exactly one of them. (Such a set of edges specifies the desired pairing.) In Figure 1(b), we show a perfect matching in the bipartite graph from Figure 1(a), with the edges joining matched pairs indicated as darker lines. So if a bipartite graph has a …