Illustrator :: Create Cut-path ONLY On Outer Edge Of Vector Image?

Here I have a vector image. The pink stroke is a cut path that the software I use at work recognises and cuts. (Used for vinyl decals/stickers etc). All the lines highlighted in pink is useless to me, as it would cut out each individual line.
 
All I need is the outer edge of the entire graphic.
 
Note my crudely drawn red lines that border the image, that is exactly what I need. As you can see, the paths run through the image. BUT...When I try to erase the paths using add anchor point then cut path at selected anchor points so I can remove that particular segment, the colour essentially "leaks" as it isn't a closed area anymore. Rasterizing the image destroys the quality and I would like to be able to perform this task without loss of quality.