1=1=1
God The Father formed his Word (the Son Jesus Christ) by his Breath (Spirit).
To the point of if they depend on each other in order for God to exist, no, God the Father is greatest of all and his existence does not depend on anything or anyone because he is the Deity. Theoretically God the Father did not have to create or form Creation, he could in theory have held his peace and not spoken his Word or breathed out his Breath, but even in this case his Word and Breath would have still dwelt inside of him, just as they have from before the beginning (John 1:1-3). Rather than God depending on the Son and Spirit, it is rather Creation that depends on all three to exist.
Realistically though since Creation does indeed exist, Creation depends on God the Father both as himself, and depends on him breathing out his Spirit and speaking his Word to form the creation. A good scriptural foundation for this is Genesis 1 and John 1.
That does not mean that God the Father depends on the Son or the Spirit though for as the Son himself says "My Father, who has given them unto me, is greater than all." (John 10:29). While the Father is greater than all and does not by necessity depend upon his Word that does not make the Trinity as fractions or separate entities, but as equals and wholly united for as the Son the Lord Jesus Christ goes on to say in the same chapter, he and his Father are one (John 10:30). Also because God the Father did ultimately decide to create Creation by speaking it into existence as recorded in the Torah, his Word and him are indeed one and they do exist and work together being one and the same (John 14). Praise the Lord God Almighty!