How to efficiently query a directed acyclic graph

I'm using mysql for one on my web applications. The application table contains a supervisor table and an employee table. employee table contains information about each employee. The supervisor table contains two columns as follows.

supervisor_id -> which is employee id of the supervisor
subordinate_id -> which is the employee id of the subordinate. 

Each subordinate can have multiple supervisors and one supervisors subordinates can be a supervisor of of some other employee. So the table records can be ass follows.

supervisor_id | subordinate_id
1             | 2
1             | 3
2             | 4
4             | 5
3             | 6
3             | 4

In the above example there is a supervisor chain. Supervisor 1 has 2, 3, 4, 5 and 6 as his subordinates. Supervisor 2 has 4, 5 as subordinates. And also it can have multiple supervisors for a subordinate.

When I querying for all subordinates for supervisor 2 currently I use a query like following.

public function getSubordinate($id) {
 $query = "SELECT * FROM supervisor WHERE subordinate_id = $id";
 // get results and return
}

So what I'm currently do is first send the id as 2 to get its immediate subordinates. Then for each and every resulting subordinates I run the query again and again to get the full subordinate chain.

This is okay with small set of data. But this supervisor table will have thousands of data so I have to do thousands of queries to find the supervisor chain and it takes times to give results.

Since subordinates can have multiple supervisors the nested set will not be a exact answer for this.

I went through this solution also . http://www.codeproject.com/Articles/22824/A-Model-to-Represent-Directed-Acyclic-Graphs-DAG-o

But