TITLE: Graph embedding that locally minimizes total edge lengths
QUESTION [7 upvotes]: I consider a graph $G$ (possibly infinite, but locally finite) embedded in the Euclidean plane $\mathbb{E}^2 \cup \{\infty\}$ such that each local perturbation of the embedding "increases the total length".  That is, for any sufficiently small neighborhood $U \subset \mathbb{E}^2$, the total length of $G \cap U$, if not empty, is minimized subject to fixed boundary points $G \cap \partial U$.
So all edges must be straight.  If a vertex has degree 3, then it must be the Fermat point of its neighbors.  If a vertex has degree 4, the adjacent edges must form two collinear pairs.  In general, the outgoing unit vectors along the edges around a vertex must sum up to 0.  I consider such embeddings as an analogy to minimal surfaces.  Examples include, of course, infinite Steiner trees --- or finite Steiner trees if I use the point at infinity wisely.
I failed to find reasonable literature about such embeddings, hence would like to ask for references.  In particular, I wonder if such an embedding may appear as (a nice supgraph of) the graph of some quasi-crystallographic tiling.

REPLY [2 votes]: see
https://arxiv.org/pdf/1803.03728.pdf ,
https://arxiv.org/pdf/1904.00483.pdf and
https://arxiv.org/pdf/1902.07872.pdf .
Here it is called geodesic nets.