Lobe, Edge, and Arc Transitivity of Graphs of Connectivity 1

Abstract

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.