Question

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!

Answer 1

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.

Answer 2

What if you take $5$ paths with $k$ vertices each and glue them together at the endpoints? So $5(k-2)+2=5k-7$ vertices in all, and circumference $2(k-1)=2k-2$.