Disconnecting a complete graph by removing edges randomly Given a complete graph with $n$ nodes, I remove edges randomly with probability&nbsp;$p$ such that I want to disconnect the graph. 

I want to find out the minimum number of edges that I must remove randomly to disconnect the graph. Also the value of $p$ where I can disconnect the graph by removing the least number of edges.
For your first question, an empty graph becomes connected roughly when $n\log n$ random edges are inserted into it, in a way quantified in Erdős and Rényi's fundamental paper On the evolution of random graphs, and recounted in textbooks on random graphs.

Your second question is not stated precisely and so is difficult to answer.