What is the relationship between cost centre and profit centre?

A Cost Center is responsible only for the incurrence of costs. A cost center does not earn any revenue and therefore generates no profit. An equipment maintenance department or an internal ac-counting department is example of a cost center

A Profit Center is a department responsible for both revenues and expenses. A department within a store, such as the hardware department, is an example of a profit center because it has both reve-nues and cost of goods sold. Because a profit center is responsible for both costs and revenues, the manager of a profit center should be evaluated on both costs and revenues generated by the profit center.

So the cost center involved in the profit center

Cost center is not directly involved in core activities of the business, however, profit center do.

you can also say that every profit center is a cost center but every cost center is not a profit center.

A cost center is a subunit of a company that is responsible only for its costs. Example of cost centers are the production departments and the service departments within a factory and administrative departments such as IT and accounting.A profit center is a subunit of a company that is responsible for revenues and costs. Often a division of a company is a profit center because it has control over its revenues, costs, and the resulting profits.Cost centers and profit centers are usually associated with planning and control in a decentralized company.

A profit centre has an income from outside the business.  A cost centre does not.  Both incur costs.

These terms are sometimes useful, but must always be used with care as they can be dangerous.  This is a generic problem with labels: labels are useful and sometimes necessary, but they can acquire meanings they should not have.

For example, cost centres are essential to the running of the business.  The more effective they are, the better the business will run.  This is not the same as being efficient. (Although efficiency is usually good.) 

Once a department is labelled as a "cost centre", there is a risk that it will be run for efficiency (to reduce costs) rather than effectiveness (to increase profits in other departments).