There are many useful graph properties described as "width parameters" that show up in algorithm analysis (especially for FPT-type algorithms). The most famous example is probably treewidth, but there is also pathwidth, cliquewidth, etc.
These properties are usually derived from divide-and-conquer algorithms for the problem at hand, and they describe the width of the optimal recursion tree. My question is whether there is a general description of graph properties that can be considered width parameters, without explicit reference to a divide-and-conquer algorithm in the background. This question might be answered by a definition along the lines of: a function $W$ from the set of finite graphs to $\mathbb{N}$ is a width parameter if it satisfies [some list of axioms]. These axioms might include things like $W(\emptyset)=0$ or that $G' \subseteq G \implies W(G') \le W(G)$, but others are likely needed too. Treewidth and friends would each correspond a different choice of $W$ satisfying the axioms.
Is there a published definition along these lines? Or is it easy to write down such a definition once we have proper intuition about what width parameters represent?