Conversation so far:
what was significant about the Minotaur's birth?
After he ascended the throne of the island of Crete, Minos competed with his brothers to rule.
did Minos beat his brothers and become ruler?
After he ascended the throne of the island of Crete, Minos competed with his brothers to rule.

Question: what was the competition?
Self-contained rewrite: what was the competition between Minos and his brothers?