Is there an efficient (possibly probabilistic/approximate) algorithm for determining whether a particular graph is the subgraph of an infinite two dimensional triangular lattice? How about three dimensional?

For context, we are interested in the question whether a graph can be geometrically embedded in 2d with fixed edge lengths, with new edges formed whenever two vertices come close enough to each other. The above combinatorial problem seems to approximate this behaviour well enough for our purposes.

P.S.: I apologize if this question is too elementary, coming from a different field I have so far failed to find the correct way to phrase it to google. Various results on degree constrained subgraphs seem the most relevant, but don't quite get to the very specific case here.

Is there an efficient (possibly probabilistic/approximate) algorithm for determining whether a particular graph is the subgraph of an infinite two dimensional triangular lattice? How about three dimensional?

For context, we are interested in the question whether a graph can be geometrically embedded in 2d with fixed edge lengths, with new edges formed whenever two vertices come close enough to each other. The above combinatorial problem seems to approximate this behaviour well enough for our purposes.

P.S.: I apologize if this question is too elementary, coming from a different field I have so far failed to find the correct way to phrase it to google.