euler cycles with biconnected components

Question

if a graph has a Euler cycle do the biconnected components have Euler cycles as well?

Answer 1

Yes. If you start with a Euler cycle for the graph and restrict to a biconnected component, then what you have is still a cycle on the biconnected component (basically, if the euler cycle leaves vertex v in the biconnected component, then you know it must return to the biconnected component through v, otherwise we could enlarge our biconnected component - contradicting its maximality).