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?



A minimally 2-connected graph H and a 3-connected graph H' both contain G, does H' contain H?
Fedja's comment definitely earns the grant. A minimal example is to let G be a star with four leaves {1,2,3,4}, let H be the bowtie obtained from G be adding the edges 13 and 24, and let H' be the wheel obtained from G by adding the cycle 1234.


