According to Orthodox trinitarianism, can the Father exist without the Son or Holy Spirit? According to the answer at Is the Eastern Orthodox doctrine of &quot;the Monarchy of the Father&quot; consistent with the Athanasian creed?, the Revised Catechism says...

094 What difference is there between the Persons of the Holy Trinity?
God the Father is neither begotten nor proceeds from any other Person, he is the cause, source and principle. This is called the Monarchy of the Father.
etc.

Does this this mean the father does not require the son or holy spirit to exist, but the son and holy spirit are dependent on the father for their existence?
Your statement is correct in principle, but the vocabulary could lead to some incorrect conclusions. 


  the father does not require the son or holy spirit to exist 


could be understood to mean that at some time the Father could have existed (or possibly even did exist) without the Son or the Spirit existing. This is not correct as the Son and the Spirit are co-eternal and co-uncreated with the Father. 

Likewise, 


  the son and holy spirit are dependent on the father for their existence


is true as the Father is the source of the Trinity. However we must not understand 'dependent' to mean the Son and the Spirit are in someway conditional -- that if things had gone differently there could have been (for example) a Father, Sister, and Heartbeat instead.

The Father is the Fountainhead of the Holy Trinity, and it is not possible for there to be a Fountainhead without that which flows from it, the Son begotten and the Spirit proceeding.