# complexity of games with tokens and only-pay-once toll roads

I was reading http://arxiv.org/abs/1201.4995, and thought back to a game I used to play,
which is close to being covered by part (c) of "Metatheorem 2" on page 4 of that paper.
(The difference is that the game's version of toll roads only charge for the first passage.)

What is the computational complexity of the following problem:

For a 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.)

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I am aware of cstheory.stackexchange.com; I asked here because I think $\hspace{2.2 in}$ this question is too low level for that site. $\:$ –  Ricky Demer Jan 11 '13 at 7:27
But there is cs.stackexchange.com now... –  ˈjuː.zɚ79365 Jun 25 '13 at 11:19