I was thinking about 2-edge connected graphs, connected graphs that cannot be disconnected without removing at least two edges. I was thinking about a slight variant on this problem - what sorts of graphs are initially connected, connected after removing any one arbitrary edge, but guaranteed to be disconnected after removing any two arbitrary edges?
Any cycle graph with at least three nodes has this property, and I can't seem to think of any other graphs that would also have the same property. Are the cycle graphs with at least three nodes the only graphs that have this property? If so, how would I go about proving this? If not, is there another class of graphs with this same property?