(This essentially copies my unaswnered question
I was reading http://arxiv.org/abs/1201.4995, and I thought
back to a game I used to play, which is close to being covered
by Metatheorem 3 (on page 5), but does not have one-way paths.
What is the computational complexity of the following problem?
For an undirected graph G whose vertices have non-negative integer weights,
for vertices s and t, is there a path from s to t such that the sum of the weights
of the vertices (including s) reached at any given point along it is always
greater than the number of distinct edges traversed to get to that point?
(The vertices are not counted with multiplicity either.)