Question

Graph embeddings with edges of equal length

Answer 1

Look up “unit-distance graphs.” The discussion is about 2D by default, but higher dimensions are also considered. Determining whether a graph is unit-distance in n dimensions for any particular n > 1 is NP-hard.

Answer 2

1) I'm assuming by graph, you mean in the sense of graph theory?



2) We should restrict ourselves to some definitions. We know what an embedding in 2D euclidean space would be as we have the definition of planarity to restrict ourselves to. What constitutes a valid embedding in 3D euclidean space in the first place?