Abstract
A connected graph X is said to be super line connected if every minimum edge cut is the set of edges incident with some vertex. It is proved that a connected Abelian Cayley graph is super line connected unless it is a cycle or the product of complete graphs, one of size 2 and one of size $m\geqq 3$. This result generalizes previous characterizations for circulants by Boesch and Wang and for hypercubes by Boesch, Lee, Wang, and Yang. A simple algebraic characterization for strongly connected Cayley digraphs that are not super arc connected and a structural characterization for the Abelian case are given. In a previous work, one of the authors presented a characterization of the connected vertex-transitive graphs that are not super line connected. The final result of the present paper generalizes this result to directed graphs.
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