I am looking for a name for this problem or any leads on an algorithm or source code:
Example: You want to find the best route to visit the 100 largest cities in the US (classic TSP) but before you can visit any given city you must visit the capital of the state it is in.
Example: You are collecting permission slips from the students of several professors. You need to visit every student and every professor but you can't visit a professor until you have seen all of his students.
Some Googling turns up the sequential ordering problem or "SOP" but there is not so much literature that I am convinced that this is a widely accepted name.
I don't think these partial orderings can be represented within the classic TSP simply by choosing which edges to use in the graph (e.g. you can't initially go from New York to Chicago, but once you visit Springfield you can) or weights, but I may be wrong.
The Sequential Ordering Problem was first introduced by Escudero in 1988 in a paper entitled "An Inexact Algorithm for the Sequential Ordering Problem" (this appeared in the European Journal of Operational Research), so this is the original name for the problem. The abstract of the paper reads:
Given the directed G= (N, A) and the penalty matrix C, the Sequential Ordering Problem (hereafter, SOP) consists of finding the permutation of the nodes from the set N, such that it minimizes a C-based function and does not violate the precedence relationships given by the set A. Strong sufficient conditions for the infeasibility of a SOP's instance are imbedded in a procedure for the SOP's preprocessing. Note that it is one of the key steps in any algorithm that attempts to solve SOP. By dropping the constraints related to the precedence relationships, SOP can be converted in the classical Asymmetric Traveling Salesman Problem (hereafter, ATSP). The algorithm obtains (hopefully) satisfactory solutions by modifying the optimal solution to the related Assignment Problem (hereafter, AP) if it is not a Feasible Sequential Ordering (hereafter, FSO). The new solution 'patches' the subtours (if any) giving preference to the patches with zero reduced cost in the linking arcs. The AP-based lower bound on the optimal solution to ATSP is tightened by using some of the procedures given in [1]. In any case, a local search for improving the initial FSO is performed; it uses 3- and 4-change based procedures. Computational results on a broad set of cases are reported.
Escudero and his collaborators have a number of papers on the subject, with references to even more. Searching for papers by him or that reference this paper may help you if you're looking through the literature.
SOP is a well-studied constrained version of the Asymmetric Travelling Salesman Problem, so much of the literature on ATSP may be related.