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, vertextransitive, primitive, regular, Cayley, and Frobenius had been previously characterized in the literature.
ASJC Scopus subject areas
- Theoretical Computer Science
- Algebra and Number Theory
- Geometry and Topology
- Discrete Mathematics and Combinatorics