Thanks for the editorial. Really builds upon intuition.
Having said that, here is my solution and TLDR:
Step 1: Does the graph have a cycle? If not print -1.
Step 2: Bridges are irrelevant. So, remove all bridges. Then each and every edge will be part of a cycle.
Step 3: Find out the vertex with most edges. If there are multiple, choose the smallest such vertex
Step 4: Delete the vertex and remove all its edges.
Step 5: Does graph still have a cycle? If so, print -1. If not, print vertex number.
Solution: CodeChef: Practical coding for everyone