I have proved a certain result for all 2-connected graphs apart from those that fit the following criteria:


1. They are "minimally 2-connected", that is, deleting any vertex will produce a graph which is no longer 2-connected, and
2. They have circumference less than $\frac{n+2}{2}$, where $n$ is the number of vertices.


I have not been able to come up with an example of such a graph. Can anyone help?


Of course the best possible outcome would be that they do not exist!



Does this graph exist?
There are lots of examples. For $t>5$, let $P\_{1},..., P\_{t}$ be internally disjoint paths with length $3$ such that each path has the vertices $x$ and $y$ as endpoints.


