Connection matrices

Suppose you want to evaluate a graph parameter on a graph G. There is a cut of size k in a graph, and while you know everything about one side of the cut, you have to pay for information about the other side. How much information do you need about the other side? To avoid the trivial solution “just tell me the value of the parameter, if my side looks like this”, let us assume that the information about the other side must be independent from what is on our side, and it is encoded in the form of an m-tuple of real numbers. Further, the answer is obtained by taking an appropriate linear combination of the numbers given, with coefficients that depend only on the graph on our side. As an example, let k = 1 (so we have a cutset {v} with one node), and suppose that we want to compute the number of independent sets in the whole graph. Then we need to know two data about the other side: the number a0 of independent sets not containing v, and the number a1 of independent sets containing v. We determine the analogous numbers b0, b1 for our side, and then the number of independent sets in the whole graph is a0b0 + a1b1. Given a graph parameter, the minimum number m = m(k) of real numbers we must know about the other side in the above scheme can be characterized as the rank of a matrix, the connection matrix. Other properties of this matrix, like whether or not it is semidefinite, also turn out to have graph theoretic significance. In this paper we survey some of these properties.