Suppose I have a directed simple graph $\Gamma$ (no edge loops or multi-edges) and a directed path $v_0v_1\cdots v_k$ joining vertex $s=v_0$ to vertex $t=v_k$. By directed path I mean that each pair $v_{i-1}v_i$ is a directed edge. Now suppose that for some $i$, the pair $v_{i-1}v_{i+1}$ is an edge, then we can omit vertex $v_i$ and still have a directed path joining $s$ and $t$.
One can think of these two paths as two routes around a triangle. Using the language of algebraic topology I would say that the two paths are homotopic. (Actually the term homotopic would mean the equivalence relation generated by such changes of path). One can then talk about homotopy classes of paths joining $s$ and $t$.
1) Does this have a name in graph theory?
and
2) Is there a theory which provides an algorithm to find the shortest path (in a weighted directed graph) in a given homotopy class?
So for question 2) one may be given a path and asked to find the shortest equivalent path. Dijkstra's algorithm applied directly wont help as it finds the shortest of all paths. It's possible Question 2) may have been considered only for graphs with no directed cycles.