Dfs algorithm and cycles question

Question

Is it true or false that for running a dfs on an undirected graph G with a simple cycle than this cycle will have exactly one back edge?

Looks to me likes its true ,is it?

Answer 1

No, it is not true. Consider the graph with vertices 1, 2, 3, 4, 5, 6 and edges 12, 23, 34, 45, 46, 56, 61. The dfs starting at vertex 1 yields two back egdes for the cycle (1,2,3,4,6).