Finite-type invariants for graphs and graph reconstructions

Many of the fundamental open problems in graph theory have the following general form: How much information does one need to know about a graph G in order to determine G uniquely. In this article we investigate a new approach to this sort of problem motivated by the notion of a finite-type invariant, recently introduced in the study of knots. We introduce the concepts of vertex-finite-type invariants of graphs, and edge-finite-type invariants of graphs, and show that these sets of functions have surprising algebraic properties. The study of these invariants is intimately related with the classical vertex- and edge-reconstruction conjectures, and we demonstrate that the algebraic properties of the finite-type invariants lead immediately to some of the fundamental results in graph reconstruction theory.