I would like to know whether it is possible to post "topics for Discussion" on Facebook Groups by using either the old Rest API or Graph API? I somehow was unable to find any function within the API which would help me to do so.
Any work around or alternate solution to the above problem is welcome.
You should be able to use the instructions in Publishing to Facebook to do this. Facebook groups appear to respond to the same requests as personal profiles do, including the /feed endpoint. Try POSTing to a group's https://graph.facebook.com/GROUP_ID/feed endpoint with the message parameter set.