  We raise some questions about graph polynomials, highlighting concepts and
phenomena that may merit consideration in the development of a general theory.
Our questions are mainly of three types: When do graph polynomials have
reduction relations (simple linear recursions based on local operations),
perhaps in a wider class of combinatorial objects? How many levels of reduction
relations does a graph polynomial need in order to express it in terms of
trivial base cases? For a graph polynomial, how are properties such as
equivalence and factorisation reflected in the structure of a graph? We
illustrate our discussion with a variety of graph polynomials and other
invariants. This leads us to reflect on the historical origins of graph
polynomials. We also introduce some new polynomials based on partial colourings
of graphs and establish some of their basic properties.
