is there such a thing?

The search term you want is KVM. A KVM allows you to switch keyboard-video-mouse (and with some models USB and audio) from multiple inputs to a single output.

In your case you want one which supports DVI and then only make use of that connection not the other ports.
 
In case you need to be using dual-link DVI and not normal DVI, I'd double check that first. On first glance those switches seem to be dual link, but best to make sure first. If you need it, that is. Doesn't sound like you do. It's for running 2560x1600 on large screens.
 
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