Does there exists a connected graph G which is a subgraph of two graphs H and H' for which
- G, H and H' have the same vertex set,
- H is minimally 2-connected (i.e. deleting any edge from H makes is not 2-connected),
- H' is 3-connected and
- H is not a subgraph of H'?
I arrived at this question when thinking about how grant applications with "anti-terrorism" in the title seem to get much more funding than those without. I came out with this problem: A terrorist wants to disconnect G by deleting edges. To prepare for this attack, I want to add edges to G to make it 2-connected (i.e. H) at minimal cost, while at the same time preparing to make it 3-connected (i.e. H') at a later date. Does the choice of H matter?