We consider optimization problems on weighted graphs where vertex and edge weights are polynomial functions of a parameter lambda. We show that, if a problem satisfies certain regularity properties and the underlying graph has bounded tree-width, the number of changes in the optimum solution is polynomially bounded. We also show that the description of the sequence of optimum solutions can be constructed in polynomial time and that certain parametric search problems can be solved in O(n log n) time, where n is the number of vertices in the graph.