There is no specific vaccinations that is mandatory for Slovakia.
However if you take a step further in being careful you may look in following, and I reccomend you do at least first 2 of them:
Hepatitis A, B
You will find this vaccination very useful wherever you travel, so just YES do it and het get vaccinated.
Tetanus / Diphtheria
You may have scratch, cut, or anything wound and this vaccination helps you in such situations. It is such fundamental one, that if you go to hospital with some wounds, they almost immediately give you injection against tetanus.
I know many people who do not have it, actually I do not know anyone who would have this vaccination.
Still it is on recommendation list
There is not much rabbies around Slovakia, but it can always happen, never say never.
In general, without this vaccination you can easily travel in Slovakia.
Some of the pages you can check further:
Health information for travellers to Slovakia | vaccination | vaccines | vaccinations | hepatitis | tetanus | diphtheria
Please note: this is not a medical advise from medical doctor, We are not reliable for any of your consequences. You always rely on your decision, based on information you can collect.