Question $1$:

$1.$ Given a DFS tree, give an algorithm that finds a shortest cycle $($cycle with least number of edges$)$ among all cycles containing exactly one backward edge.

$2.$ Is this cycle necessarily the shortest cycle in the graph?

Remarks: All the graphs are without self loops and parallel edges, and anti$-$parallel edges. In all the algorithms, always explain their correctness and analyze their complexity. The complexity should be as small as possible. A correct algorithm with large complexity, may not get full credit