The question needs to be broken down, we need to know precisely what is being asked in order to answer it.
The three persons of the Godhead are not dependent on each other for existence. Each person of the Godhead is God. And the fundamental truth about God is that he is eternal: "From eternity to eternity I am God. No one can snatch anyone out of my hand. No one can undo what I have done", Isaiah 43:13. God cannot not exist. Each person of the Godhead is the almighty, ever-existing, "I am". The Holy Spirit cannot not exist, the Son cannot not exist, the Father cannot not exist, and only these three cannot not exist.
Things start to get a bit "tricky" after this, and we have to hang on to every word of the Holy Bible. The real question might be this: "Is the Bible truly the word of the Living God?" If it is, then it doesn't matter how illogical it may seem to us.. we must believe it. The person who has not been willing to prayerfully give the Bible a fair hearing is really in no position to decide if it is true.. if they say it is against logic, then who is arguing? If they say it cannot be true because it teaches things that sound illogical, the Christian would say "All I know is once I was blind but now I can see", John 9:25, there has been a miraculous change in me, and it is because of what the word of God has taught me to believe. The Christian's experience is just as a stanza of a hymn tells it:
"Run, John, and work, the Law commands, Yet finds me neither feet nor hands;
But sweeter news the Gospel brings; It bids me fly, and lends me wings."
I know the Bible is the true word of God, even if I do not, and maybe cannot, understand everything it says. It claims to be God's word: it says "All scripture is given by inspiration of God", 2 Timothy 3:16; 815 times it uses the phrase "saith the LORD", (KJV); no one should add anything to it or take away anything from it, Revelation 22:18-19. If it is not the word of God then it is the most terrible, terrible lie.. and who can believe that who has read it? (Muslims are often afraid to read the Bible.. that is the problem. What they have been taught has made them fearful of having an open mind, fearful of being converted, fearful of how other Muslims would treat them.)
The Bible tells us there are three persons in the Godhead, and the three persons of the Godhead are one God.
Jesus is God: the Apostle Thomas said to Jesus "Ho Kurios mou kai ho Theos mou", "The Lord of me and the God of me", John 20:28, https://www.scripture4all.org/OnlineInterlinear/NTpdf/joh20.pdf; and "God was manifest in the flesh", 1 Timothy 3:16, See "The Revision Revised" by John W. Burgon.
The Holy Spirit, who the Israelites provoked 40 years in the wilderness, is the living God, Hebrews 3:7-12.
We are told the three persons of the Godhead were involved in both the work of creation of all things, Genesis 1:1-3; and in the work of salvation of God's people, Matthew 28:19, John 14:26.
In creation, The Father gave the Word, (which is our Lord Jesus Christ, the Word of God), and spoke the universe into existence, and the Holy Spirit was hovering over the waters, Genesis 1:2, enabling the work(?), showing minute care over the details(?). Did they depend on each other for the work of creation? Yes. Could they have made the universe so that only one was making without the other two? I guess anyone of the three persons could. But we worship God in all his ways: we worship God for who he is, what he has done, and how he has chosen to do it, as he has revealed his being and his ways in his own word the Bible, one God in three persons.
The same interaction is found in the work of salvation. The Father sent His only begotten Son to live a sinless life on our behalf, the only man who has ever been able to live without sin. He then died for our sins, a perfectly sinless sacrifice, and then rose from the dead because "it was not possible for death to keep its hold on him", Acts 2:24. He "abolished death" for his people, 2 Timothy 1:10, and by his resurrection he was emphatically "declared to be the Son of God", Romans 1:4. The Holy Spirit worked in our hearts, to persuade us of our own helplessness in sin, and to enable us to believe on the Son for salvation, and to come to God the Father through the Son. By the aid of the Spirit we are saved. Could God have brought salvation some other way? I guess so. Maybe the Father could have come himself(?). But I don't like to even think about other ways, I love the way the Father sent his one and only Son, our Lord Jesus Christ, who not only died for us, but also, as a man, showed us how to worship the Father, with the aid of his Holy Spirit.
All those who believe God exists in only one person have a slight problem. Our Lord Jesus once said to another Jew something like "the whole duty of man is to love your neighbour as you love yourself, and to love God with all your heart. This sums up the law the prophets," Matthew 22:40. It is God-like to do this: God loves us in this way.
Now, very near the beginning of the Bible it says "It is not good for man to be alone", Genesis 2:18. But why? It is because man is created "in the image of God", Genesis 1:26-27. It is not good for man to be alone: Man needs someone, someone with the same nature, to love: he is not able to be fully complete without having someone to love: this is because man is in the likeness of God.
"God is love", 1 John 4:16. And because God is love, there is more than one person in the Godhead. Before God created anything, the three persons of the Godhead existed in perfect happiness and contentment. They did not need to create the Universe to be perfectly fulfilled. Each person of the Godhead loved the other two. Before the Universe was created, for an eternity, the three persons of the Godhead were entirely fulfilled in loving each other, in the contemplation of the perfection of the other two, and in each other's love.
The three persons of the Godhead are mutually dependant in love.