This chapter gives an overview of local graph operations, recursions and reductions used in the computation of graph polynomials. It also gives a list of new results, including new local graph operations and their applications, recursive relations for known graph polynomials, the definition of more general graph polynomials that allow the derivation of new recursions for the domination, edge cover, acyclic, and covered component polynomial. Graph polynomials are considered as generating functions for some sequences of numbers of certain (induced, spanning, or general) subgraphs of a graph. A local graph operation assigns to any graph G another graph H such that G and H differ only in the neighborhood of a vertex or an edge. Local graph operations occur in recursive relations for graph polynomials. They are also the basis for graph reductions.