# complexity of games with just doors and keys

(This essentially copies my unaswnered question
from math.stackexchange.com/questions/275685)

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.)

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If I understand your question correctly, by path you mean that we build up a tree starting from s by always taking an edge adjacent to our current tree, right? –  domotorp Jan 19 '13 at 9:32
Yes. $\:$ (I suppose I should have said "walk".) $\;\;$ –  Ricky Demer Jan 20 '13 at 2:35