Contractors and connectors of graph algebras

We study generalizations of the “contraction-deletion” relation of the Tutte polynomial, and other similar simple operations, to other graph parameters. The question can be set in the framework of graph algebras introduced by Freedman, Lovasz and Schrijve, and it relates to their behavior under basic graph operations like contraction and subdivision. Graph algebras were introduced in the same paper to study and characterize homomorphism functions. We prove that for homomorphism functions, these graph algebras have special elements called “contractors” and “connectors”. This gives a new characterization of homomorphism functions.