Graph Theory Cycles in Nonseparable graphs Let $k$ be the maximum length of a cycle in a nonseparable graph $G$. Prove that if $C$ and $C'$ are any two $k$-cycles in $G$, then $C$ and $C'$ have at least two vertices in common.


Nonseparable meaning biconnected. Does anyone have a hint?
Theorem. If the graph G has a vertex v that is connected to a vertex of the component G1 of G, then v is also a vertex of G1.


Suppose there exists one and only edge connecting C to C'. Removing this edge would yield two connected components.

Suppose there exists two edges connecting C to C'. Removing one edge would yield one connected component. C and C' have at least two vertices in common.
Theorem. If the graph G has a vertex v that is connected to a vertex of the component G1 of G, then v is also a vertex of G1.


Suppose there exists one and only edge connecting C to C'. Removing this edge would yield two connected components.

Suppose there exists two edges connecting C to C'. Removing one edge would yield one connected component. C and C' have at least two vertices in common.