Untangling a graph Assume you have a 4-valent graph (i.e., a knot universe, i.e. a collection
of self-intersecting curves). Your allowed moves are the equivalents
of Reidemeister 1, 2, 3, just with 4-nodes instead of over/underpasses.
(I don't think you need a pic :-) 
Is the following logic OK? Effectively, your graph is a knot where
overpasses can be turned into underpasses ad lib. Since this is an 
unknotting move, and the rule set above is equivalent to the "real" Reidemeister 
plus some crossing flips, the pseudo-Reidemeister moves can turn any graph into 
disconnected loops.
(If yes, I think I have a complete no-brainer proof for the invariance
of the Dubrovnik 2-variable polynomial, but I bet you find loads of brains
lacking in the fine print :-) BTW, is there a no-brainer proof for the
fact that crossing flip is an unknotting move? (Or maybe the graph version
is even simpler to prove.) Obviously, you as knot theory profis don't
actually need another proof for the Dubrovnik, since one exists...
but I prefer one my no-brain can understand :-)
 A: The comments here seem to be answers and the answers comments, so I do not know if you feel that your question has been answered or not!  At the risk of beating a dead horse, let me make sure. I am assuming you mean a quadravalent graph in the plane, so there are no ordinary crossings. Following the algorithm of the first commenter (traverse the graph, and mark the first arc you go over in each crossing as nominally the "over" strand), you can identify the graph with an unknot with an overlapping projection.  The sequence of Reidemeister moves that turns that unknot into the standard projection of the unknot give you a sequence of your Rmoves that does the same thing.
In answer to the first answer, I believe the 4-valent version of the Reidemeister moves means draw the ordinary Reidemeister moves and the replace each crossing with a 4-valent vertex.  Anyway, that was the assumption of my answer!
A: This is a comment requesting clarification, not an answer.
At least to me, it is not self-evident what is the 4-valent equivalent of the three Reidemeister
moves.  For example, Reidemeister I seems fundamentally 3-valent.
If you allow the move shown below, which is akin to Reidemeister II, and if your collection
of curves includes only proper crossings, as illustrated, then indeed
the graph can be converted to a collection of disjoint (but often nested) loops:
in the example, four loops nested inside a fifth.
       
Perhaps you could clarify your intent?
