If your daughter was born (In England or Wales) after 1st December 2003 the fact that her father's name is on her birth certificate automatically gives him parental responsibility.
If you've not got a residence order you will be committing a serious criminal offence if you try to take your daughter out of the country (even for a day trip) without the consent of her father.
If you've got a residence order you can take your daughter out of the country for up to one month without her father's permission but, again, you'll be committing a serious criminal offence if you try to take her out of the country for longer than one month without his consent. (Further, if you did so, your daughter's father could seek an order from the Slovak courts -under international agreements on child abduction - to have her returned to the UK).
You MUST obtain the father's (written) permission in order to return to Slovakia permanently with your daughter.
http://www.direct.gov...ildpassport/DG_174106
Chris