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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   Λ   and a “code” assigned to each orbit of   Aut(Λ)  , there exists a unique lobe-transitive graph   Γ   of connectivity 1 whose lobes are copies of   Λ   and is consistent with the given code at every vertex of   Γ  . 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.

Lobe, edge, and arc transitivity of graphs of connectivity 1