Breaking Apart Path Object

Hello all, I have a basic Illustrator question which I am sure has a very simple answer that is alluding me.

I have the task of using an EPS created by someone else, and have to apply a stroke to it.

Now, the EPS is of a logo image, made up of multiple paths. Now, the final logo ends up being one enclosed shape.

Now, my problem is that the paths are a collection of enclosed "shapes" and if I apply a stroke to the paths the result will be a stroke around the shape (which is great), PLUS a few horizontal lines because the mini shapes are enclosed.

One way I can explain it is if you were to make a square through the creation of four equally sized sub-squares. If you apply a stroke to all of the squares, you will get your "border" but also 4 interior lines.

So, I want to know how I can go about disconnecting or breaking apart two anchor points, so that they are no longer joined. I assume if you have a curved line you can still apply a stroke to it.

Anyway, thanks!

Because my explanation is admittedly weak, here is an example of what I am talking about.

The image included below is one that could be considered to be made up of two paths, the top and bottom shapes, outlined in red.

Now, if I apply a stroke to both paths, it will include the red diagonal line because it is the edge of each path.

I would like to be able to eliminate the line between these points (technically in this case, two lines).

Thanks (hope that helps)