This scenario is not regulated by the Slovak Labour Code. From legal point of view, this should be treated as withdrawal from the employment contract before start date. However, employees do not have such option under the Slovak Labour Code.
The employment relationship is established by the written employment contract. However, the employment relationship is legally formed (i.e. starting to have its legal effects) only as of the start date.
So, if employee decided to resign before the start date (and not to come to work on start date), there is basically no employment being formed at that time (even if contract was agreed and formally signed) and thus there is no need for making any (legal) acts towards employment relationship / employment contract in order to be terminated.
However, to avoid any doubts in the future, the notice letter must be sent to employee, in which the employer simply accepts employee’s resignation and confirms that no employment relationship has been created between the parties and no employment related rights or obligations exist between them. The notice letter must be signed in a wet ink and delivered to the employee at his/her Addreess.