GIMP :: Selecting Just Single Line For Converting Into Path

Let's say you just have a simple black line in Gimp. You then use the color select tool to select it, and you plan on then converting that Selection into a single line Path.

From:

Into:

You select the line, but the Selection tool actually travels a loop around the line. So, when you turn that Selection into a Path, you do not have a single line, but you instead have a loop.

So my question is this: is there any mechanism for making your Selection in GIMP just a single line so that it can then be converted into a Path?

Originally, I was trying to achieve this in Inkscape (and started a thread about it in the Inkscape forum). However, it appears that, although other Vector Art programs can achieve this, Inkscape cannot. So I was thinking that I could instead use Paths in GIMP instead, but this is not possible because of the nature of Selections, as far as I can tell.

Here's an attempt to illustrate this request: