Sorry for my English.
Here is the question:
Definition: Legal's up-path in a graph from s to t is existent if and only if for every Vi, Vi+1 (for each i) fulfill w(Vi)<=w(Vi+1), when w(v) is a weight of the vertex.
We know that there is an algorithm like BFS that solve the problem of finding the shorter path in a Graph. We need to find, by reduction, an algorithm that solve the problem of legal's up-path.
I try to split each vertex, but it didn’t work (for example: for Vi=4, I tried to create 4 Edge, and then activate the BFS, but it didn’t work.