The standard complexity classes of Complexity Theory do not allow for direct classification of most of the problems solved by heuristic search algorithms. The reason is that, in their standard definition, complexity classes are specifically tailored to explicit, instead of implicit, graphs of state or problem reduction spaces. But the usual practice works over implicit graphs. To allow for more precise comparisons with standard complexity classes, we introduce here a model for the analysis of algorithms on graphs given by vertex expansion procedures. It is based on previously studied concepts of ``succinct representation'' techniques, and allows us to prove PSPACE-completeness or EXPTIME-completeness of specific, natural problems on implicit graphs, such as those solved by A*, AO*, and other best-first search strategies.