The easiest way to do this is to play around, I think. One, two, or three edges don’t seem to be enough, but removing four edges gives us a straight graph on six vertices.
Note that a connected graph with no cycle is called a tree. $A$ tree on n vertices has $n-1$ edges. We need to delete edges from the given graph so that only $5$ edges remain. Hence, $4$ edges must be deleted.
