Has there been any effort to formalise the subset of NP that lends itself to SAT resolution (is there something between x^n and n^x)? For example, what are the

Has there been any effort to formalise the subset of NP that lends itself to SAT resolution (is there something between x^n and n^x)? For example, what are the defining characteristics of a graphs for which the travelling salesman problem is resolvable without resorting to brute force?