A relationship is a promise.
I am the father of my children. While my children are my own, it is not that because they were born through me that I have a relationship with them of being a father. I am their father because I choose to be in this relationship with them and honour the promise of this relationship. There is a certain set of expectations that my children, my wife, my parents, my children’s school and the society have of me as a father. And when I honour their expectations, I do truly become a father in their eyes.
My brother on the other hand has two adopted sons. They were not born through him, and yet, his promise as a father is by no means any less than mine. There are biological fathers, who do not keep the promise of being fathers. So, my claim is that being a father is not about blood, but about a promise.
Similarly, all other relationships are promises. A relationship between a client and the vendor company, a relationship between a subordinate and his line manager, a relationship between a husband and wife and so forth.
Responsibility is being willing to be the cause in the matter. It is taking the posture that you are the source or the cause of something, and that outcomes can be shifted by your actions.