I've been reading Saxe's proof that 1-embeddability of integer weighted graphs is NP-Complete (http://www.math.columbia.edu/~dpt/RigidityREU/Saxe79EmbedNPHard.pdf Theorem 3.2). I don't understand how this proof works.
First of all, does "1-embeddability" mean mapping each vertex to the real line? What is meant by a "direction" as mentioned in Theorem 3.1?
I understand that PARTITION takes a set of integers and partitions them so that when all integers of $S_1$ are added, and when all integers in $S_2$ are added, both of these sums are the same.
In the proof of Theorem 3.2, it seems to be saying that we take this set of integers, and make them into weights of a cyclic graph. So we have one big cycle, and the edge weights are the integers. After this, I'm lost.
Why is it that the two sets $S_1$ and $S_2$ as defined add to the same thing?
And this proof is using a cyclic graph, how does it extend to general graphs? would we just remove an edge from this so its not cyclic?