Converting graphs to sets of paths I have an Euclidean, undirected graph: each vertex is a point on the 2D plane, so the weight of each edge is the Euclidean distance between the vertices.


The number of vertices with no edges is small in comparison to the total number of vertices. 
No vertex has more than two edges connected to it.


How can I convert this graph to a closed shape, so each vertex will have exactly two edges connected to it and there will be a path between any two vertices? I want to do it with minimum change in the total weight of all the edges.
This should be NP-hard, here's a way you could try to do a reduction:

Take any graph $G$ embedded in the $\mathbb{R}^2$ plane. Let $c_{min} = \min_{e \in E}\{c(e)\}$  be the minimum edge weight in the graph ($c$ is the weight function of the edges of $G$, i.e. the euclidean distance). To build $G'$, split every vertex $v = (x,y)$ into $v_1 = v$ and $v_2 = (x+\epsilon,y)$ where $\epsilon = \frac{c_{min}}{2^n}$, connect $v_1$ and $v_2$ with an edge weighted by their euclidean distance. Obviously, no vertex in $G'$ is incident with more than two edges, and there are no vertices that are incident with no edges.

What you now would have to prove is something like "there's a TSP route of weight $\le w$ in $G$ iff there's a TSP route of weight $\le w + n \cdot \epsilon$ in $G'$. I don't know how to prove that, but some geometry and use of the triangle inequality might help. I expect the choice of $\epsilon$ as vanishingly small will be enough to give the reduction.

EDIT: The reduction will remain valid after your edit.