Very occasionally there is a question on math stack exchange that I can actually give some kind of useful answer to. This question caught my eye; the questioner asks whether graphs with the same degree sequence have the same biconnectivity property. Or to put it another way, are there any pairs of graphs with the same degree sequence but one is biconnected and the other is not?

I thought of an example of such a pair, and another user supplied a much simpler one, but here are both pairs:

The vertices are coloured by degree (4 = orange, 3 = blue, 2 = green); the red arrows show a transformation of the top graph into its non-biconnected partner. Obviously there may be more than one, I've no idea how many members of a set of isodegree graph might be expected to be biconnected.

For the record, here is the transformation of the bottom pair:

The red lines bisecting edges indicate a kind of 'bond breaking' although it's not meant to represent an actual chemical reaction!

I thought of an example of such a pair, and another user supplied a much simpler one, but here are both pairs:

The vertices are coloured by degree (4 = orange, 3 = blue, 2 = green); the red arrows show a transformation of the top graph into its non-biconnected partner. Obviously there may be more than one, I've no idea how many members of a set of isodegree graph might be expected to be biconnected.

For the record, here is the transformation of the bottom pair:

The red lines bisecting edges indicate a kind of 'bond breaking' although it's not meant to represent an actual chemical reaction!