More about Dogs

Whilst on the subject of dogs, I'm sure I read somewhere that the
English word 'dog' was derived from the verb 'to do', presumably
because dogs were working hounds. It has been drawn to my attention
that there is a German word 'Dogge' which is used for certain large
breeds.
Does this make the link with 'do' untenable, could the German word
have a Plattdeutsch origin or has the German verb 'tun' mutated from
something closer to the word 'do'?
Thank you in advance.
Regards
David