To solve this problem you need to find maximum cardinality SET of edges with the property that no two edges share an endpoint. It can be done using Hopcroft-Karp algorithm. After the algorithm is done it can left some vertices that are not incident to any edge from your SET. Add one edge for each of them.

Also is there anyone who solved it using Ford-Fulkerson algorithm? I personally have TL, solving it that way.

I decided using the algorithm of kuhn