  We give necessary and sufficient conditions for lobe-transitivity of locally
finite and locally countable graphs whose connectivity equals 1. We show
further that, given any biconnected graph $\Lambda$ and a "code" assigned to
each orbit of Aut($\Lambda$), there exists a unique lobe-transitive graph
$\Gamma$ of connectivity 1 whose lobes are copies of $\Lambda$ and is
consistent with the given code at every vertex of $\Gamma$. These results lead
to necessary and sufficient conditions for a graph of connectivity $1$ to be
edge-transitive and to be arc-transitive. Countable graphs of connectivity 1
the action of whose automorphism groups is, respectively, vertex-transitive,
primitive, regular, Cayley, and Frobenius had been previously characterized in
the literature.
