Suppose I have planar graph with a fixed number of vertices and edges. Each vertex must be located at a unique integral point on the unit plane ( $(0,0), (1,1), (1,0)$, etc), and each edge must be a straight line segment. Any number of vertices can be moved, rotated, flipped, etc as long as the graph remains planar (no intersections after movement).
Example below.
Here is a graph of six vertices, and given edges. The top image is arranged with total edge length ~ $10.656$. However, this can be improved by rearranging the vertices as in the second image. This reduces the total edge length to ~ $10.242$. (I don't know if this is minimal)
Is there an algorithm to find the minimum total edge length in such graphs? (Or an algorithm for a more restricted problem statement, e.g. only up to 4-connected vertices?)
